#72338794

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	#245233	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=ij,c^2=j,d^2=i,e^2=ij,b^a=bf,c^a=cfj,c^b=cfj,d^a=dg,d^b=dh,e^a=eg,e^b=eij,e^c=eghi,f^d=fi,f^e=fij,g^b=gij,g^c=gi,h^a=hj,h^c=hi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338795, #72338797, #72338806, …
maximal normal subgroup of	#72338795, #72338797, #72338806, …

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