#7624168

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1502 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)(28,59)(30,60)(31,47)(33,43)(35,58)(37,46)(38,51)(41,54)(44,62)(45,61)(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)(58,66)(59,69)(60,70)(61,72)(62,71);(9,12,16,30)(10,18,21,45)(11,22,26,52)(13,28,15,37)(14,33,35,62)(17,38,42,65)(19,43,36,44)(20,46,47,59)(23,49,24,50)(25,56,57,71)(27,58,60,32)(29,31,61,34)(39,63,40,64)(41,67,68,69)(48,66,70,55)(51,53,72,54);(3,8)(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)(20,32,47,58)(22,49,52,50)(25,53,57,54)(27,59,60,46)(29,33,61,62)(38,63,65,64)(41,55,68,66)(48,69,70,67)(51,56,72,71);(2,6)(5,7)(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)(20,32,47,58)(22,50,52,49)(25,54,57,53)(27,59,60,46)(29,33,61,62)(38,64,65,63)(41,66,68,55)(48,67,70,69)(51,71,72,56)

Polycyclic presentation:

2^9|c^2=i,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531364);