#72284393

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245225	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,12)(5,6)(8,13)(9,14)(10,11)(15,16)(20,28)(23,31)(24,29)(25,32);(1,3,6,11)(2,7,5,10)(4,9,13,16)(8,14,12,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,31)(30,32);(1,4)(2,8,5,12)(3,9)(6,13)(7,14,10,15)(11,16)(17,19)(18,23)(20,21,28,30)(22,27)(24,32,29,25)(26,31);(1,2,6,5)(3,10,11,7)(4,12)(8,13)(9,14)(15,16)(17,20)(18,24)(19,21,27,30)(22,28)(23,32,31,25)(26,29);(1,5,6,2)(3,10)(4,12,13,8)(7,11)(9,15)(14,16)(17,21)(18,25,26,32)(19,20,27,28)(22,30)(23,29)(24,31)

Polycyclic presentation:

2^10|b^2=i,c^2=f,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dgij,d^b=dij,d^c=dfgi,e^a=eghi,e^b=efij,e^c=egij,e^d=eh,f^a=fi,f^d=fij,f^e=fij,g^c=gij,g^e=gj,h^b=hij,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285163, #72310940, #72310975, …
maximal normal subgroup of	#72285163, #72310940, #72310975, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);