#7623849

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,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, 7531045);