#72306534

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#499832
commutator, Frattini, cosocle	C₂×#127	C₂⁴
LCS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,d^2=hj,g^2=ij,h^2=j,b^a=bij,c^a=ce,c^b=cf,d^a=di,d^b=dghij,d^c=dh,e^b=ej,e^d=ej,f^b=fh,f^d=fhj,g^c=ghj,g^f=gij,h^b=hi,h^c=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306540, #72306554, #72306757, …
maximal normal subgroup of	#72306540, #72306554, #72306757, …

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