#72279401

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#13082
commutator	C₂²×Q₈	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂⁴
socle	C₂	C₂×#13082
prime radical	#6437437	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#532 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4150 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)(3,11,13,28,15,27,14,29)(4,16)(5,19)(6,9)(7,12,23,31,22,30,24,32)(8,20)(10,18)(17,25)(21,26);(7,22)(11,27)(23,24)(28,29);(1,3)(2,7,9,22)(4,12)(5,13)(6,15)(8,23,19,24)(10,11,26,27)(14,20)(16,28,25,29)(17,30)(18,31)(21,32);(2,8)(4,17)(5,20)(7,23)(9,19)(10,16)(11,28)(12,30)(13,14)(22,24)(25,26)(27,29)

Polycyclic presentation:

2^10|a^2=g,c^2=e,f^2=j,g^2=ij,h^2=j,b^a=bij,c^a=cf,c^b=ce,d^a=dhi,e^a=ej,f^a=fh,f^d=fj,g^c=gh,g^d=gij,g^f=gij,h^a=hi,h^c=hj,h^d=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279635, #72340790, #72340862
maximal normal subgroup of	#72279635, #72340790, #72340862

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