#7623053

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60459	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530249);