#7623970

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
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1342 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:

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

Polycyclic presentation:

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

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, 7531166);