#503906

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,23)(12,30)(13,34)(14,21)(15,22)(16,41)(17,26)(18,32)(19,31)(20,39)(24,59)(25,43)(27,58)(28,53)(29,67)(33,50)(35,55)(36,52)(37,47)(38,54)(40,49)(42,64)(44,65)(45,66)(46,72)(48,69)(51,68)(56,61)(57,62)(60,70)(63,71);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,17,21,44)(12,25,32,63)(13,35,37,66)(15,29,16,28)(18,46,30,57)(19,43,51,71)(20,53,54,67)(22,55,41,45)(23,47,58,34)(24,40,61,33)(26,38,65,39)(31,64,68,60)(36,62,69,72)(42,50,70,49)(48,56,52,59);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,31)(14,32)(15,40)(16,33)(17,42)(19,47)(20,48)(21,30)(22,50)(23,56)(25,29)(26,60)(27,61)(28,63)(34,51)(35,62)(36,39)(37,68)(38,69)(41,49)(43,53)(44,70)(45,57)(46,55)(52,54)(58,59)(64,65)(66,72)(67,71);(9,12)(10,19)(11,25)(13,36)(14,32)(15,33)(16,40)(17,43)(18,47)(20,49)(21,51)(22,52)(23,57)(24,29)(26,64)(27,63)(28,61)(30,34)(31,39)(35,62)(37,69)(38,68)(41,48)(42,53)(44,71)(45,56)(46,58)(50,54)(55,59)(60,65)(66,72)(67,70)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411102);