#72284733

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3528 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=h,c^b=ch,d^a=dh,d^c=dg,e^a=ef,e^b=egh,e^c=ef,e^d=ef,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,g^e=gi,h^c=hj,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284847, #72286185, #72310940, …
maximal normal subgroup of	#72284847, #72286185, #72310940, …

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