#72251947

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#47031
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#4320282
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|c^2=fgj,d^2=fgij,e^2=j,g^2=j,h^2=j,c^b=cf,d^a=df,d^b=df,d^c=dghj,e^a=eghj,e^b=eghij,e^c=ehij,e^d=ehij,f^e=fj,g^e=gj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251951, #72251956, #72251959, …
maximal normal subgroup of	#72251951, #72251956, #72251959, …

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