#7624833

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1287 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,34)(15,36)(16,21)(18,30)(20,28)(22,45)(23,38)(24,39)(25,52)(26,41)(29,54)(31,60)(32,56)(33,44)(35,61)(37,49)(40,46)(42,58)(43,59)(47,67)(48,68)(50,69)(51,66)(53,65)(55,57)(62,71)(63,70)(64,72);(1,3)(2,5)(4,8)(6,7)(9,11,16,26)(10,17,21,41)(12,22,32,50)(13,23,15,24)(14,25,35,53)(18,37,43,64)(19,38,36,39)(20,40,44,66)(27,45,56,69)(28,46,33,51)(29,47,31,48)(30,49,59,72)(34,52,61,65)(42,62,55,63)(54,67,60,68)(57,70,58,71);(9,12,28,57,15,31,35,59,16,32,33,58,13,29,14,30)(10,18,34,54,19,42,44,56,21,43,61,60,36,55,20,27)(11,22,46,70,24,48,53,72,26,50,51,71,23,47,25,49)(17,37,52,67,38,62,66,69,41,64,65,68,39,63,40,45);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,26,24)(12,29,32,31)(14,33,35,28)(17,38,41,39)(18,42,43,55)(20,34,44,61)(22,47,50,48)(25,51,53,46)(27,54,56,60)(30,58,59,57)(37,62,64,63)(40,52,66,65)(45,67,69,68)(49,71,72,70);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,24,26,23)(12,29,32,31)(14,33,35,28)(17,39,41,38)(18,42,43,55)(20,34,44,61)(22,48,50,47)(25,46,53,51)(27,54,56,60)(30,58,59,57)(37,63,64,62)(40,65,66,52)(45,68,69,67)(49,70,72,71)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532029);