#72284473

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	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233541	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3894 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)(3,6)(4,11)(5,8)(7,13)(9,14)(10,15)(12,16)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,28)(30,32);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19)(18,20)(21,30,31,25)(22,26)(23,27,29,32)(24,28);(1,4,5,13)(2,7)(3,9,10,12)(6,14)(8,11)(15,16)(17,20)(18,19,24,26)(21,27,31,32)(22,28)(23,25)(29,30);(6,15)(9,12)(17,21)(18,23)(19,25)(20,27)(22,31)(24,29)(26,30)(28,32);(3,10)(4,13)(6,15)(7,11)(18,24)(20,28)(23,29)(27,32)

Polycyclic presentation:

2^10|b^2=f,c^2=fghj,c^a=ch,c^b=cg,d^a=dfh,d^b=dghij,d^c=dfh,e^a=eij,e^b=ei,e^c=ej,f^c=fi,f^d=fij,g^a=gj,g^b=gi,g^c=gij,g^d=gij,h^b=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285517, #72340342, …
maximal normal subgroup of	#72284709, #72285517, #72340342, …

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.6);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);