#7623124

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	#33516	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1497 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)(11,17)(12,19)(13,20)(14,33)(15,37)(16,22)(18,29)(21,35)(23,39)(24,40)(25,55)(26,59)(27,42)(28,46)(30,48)(31,47)(32,49)(34,43)(36,62)(38,51)(41,57)(44,61)(45,60)(50,66)(52,68)(53,67)(54,69)(56,63)(58,72)(64,71)(65,70);(2,5)(6,8)(9,11,16,27)(10,17,22,42)(12,23,30,52)(13,24,15,26)(14,25,36,58)(18,38,45,65)(19,39,48,68)(20,40,37,59)(21,41,49,69)(28,50,31,53)(29,51,60,70)(32,54,35,57)(33,55,62,72)(34,56,61,71)(43,63,44,64)(46,66,47,67);(9,12)(10,18)(11,23)(13,28)(14,34)(15,31)(16,30)(17,38)(19,33)(20,43)(21,47)(22,45)(24,50)(25,56)(26,53)(27,52)(29,35)(32,60)(36,61)(37,44)(39,55)(40,63)(41,67)(42,65)(46,49)(48,62)(51,57)(54,70)(58,71)(59,64)(66,69)(68,72);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,23)(13,31)(14,29)(15,28)(16,30)(17,39)(18,33)(20,47)(21,43)(22,48)(24,53)(25,51)(26,50)(27,52)(32,61)(34,35)(36,60)(37,46)(38,55)(40,67)(41,63)(42,68)(44,49)(45,62)(54,71)(56,57)(58,70)(59,66)(64,69)(65,72);(9,13,16,15)(10,20,22,37)(11,24,27,26)(12,28,30,31)(14,32,36,35)(17,40,42,59)(18,43,45,44)(19,46,48,47)(21,33,49,62)(23,50,52,53)(25,54,58,57)(29,34,60,61)(38,63,65,64)(39,66,68,67)(41,55,69,72)(51,56,70,71)

Polycyclic presentation:

2^9|b^2=i,d^2=f,e^2=i,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, 7530320);