#7623321

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1599 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)(14,32)(15,36)(16,21)(18,29)(20,34)(22,48)(23,39)(24,40)(25,55)(26,42)(28,58)(30,59)(31,47)(33,43)(35,61)(37,46)(38,51)(41,54)(44,62)(45,60)(49,69)(50,67)(52,70)(53,68)(56,63)(57,66)(64,71)(65,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,25)(15,24)(16,26)(18,38)(19,39)(20,41)(21,42)(27,48)(28,49)(29,51)(30,52)(31,53)(32,55)(33,56)(34,54)(35,57)(36,40)(37,50)(43,63)(44,64)(45,65)(46,67)(47,68)(58,69)(59,70)(60,72)(61,66)(62,71);(3,8)(5,7)(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,43)(20,46)(21,45)(23,49)(24,50)(25,56)(26,52)(27,32)(29,34)(31,60)(35,62)(36,44)(39,63)(40,64)(41,67)(42,65)(47,58)(48,55)(51,54)(53,72)(57,71)(59,61)(66,70)(68,69);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,23,26,24)(12,28,30,37)(14,31,35,34)(17,39,42,40)(18,43,45,44)(20,32,47,61)(22,49,52,50)(25,53,57,54)(27,58,59,46)(29,33,60,62)(38,63,65,64)(41,55,68,66)(48,69,70,67)(51,56,72,71);(9,13,16,15)(10,19,21,36)(11,24,26,23)(12,28,30,37)(14,31,35,34)(17,40,42,39)(18,43,45,44)(20,32,47,61)(22,50,52,49)(25,54,57,53)(27,58,59,46)(29,33,60,62)(38,64,65,63)(41,66,68,55)(48,67,70,69)(51,71,72,56)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530517);