#6584335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1941 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)(3,8)(4,6)(5,7)(9,10,14,21)(11,23,27,17)(12,19,31,48)(13,20,35,53)(15,37,16,22)(18,39,47,32)(24,57,59,45)(25,49,29,41)(26,56,62,42)(28,43,38,44)(30,50,65,71)(33,67,63,51)(34,52,66,46)(36,55,40,54)(58,69,61,70)(60,72,64,68);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,17,21,23)(12,29,31,25)(13,33,35,63)(15,28,16,38)(18,45,47,57)(19,49,48,41)(20,51,53,67)(22,43,37,44)(24,39,59,32)(26,40,62,36)(30,58,65,61)(34,64,66,60)(42,54,56,55)(46,68,52,72)(50,70,71,69);(9,12,16,39,14,31,15,32)(10,18,37,48,21,47,22,19)(11,24,38,29,27,59,28,25)(13,30,40,34,35,65,36,66)(17,41,44,57,23,49,43,45)(20,46,55,71,53,52,54,50)(26,58,33,64,62,61,63,60)(42,68,67,69,56,72,51,70);(1,4,5,3)(2,6,7,8)(9,12,14,31)(10,19,21,48)(11,25,27,29)(13,34,35,66)(15,32,16,39)(17,41,23,49)(18,22,47,37)(20,52,53,46)(24,28,59,38)(26,61,62,58)(30,40,65,36)(33,60,63,64)(42,69,56,70)(43,45,44,57)(50,54,71,55)(51,68,67,72);(9,13)(10,20)(11,26)(12,30)(14,35)(15,36)(16,40)(17,42)(18,46)(19,50)(21,53)(22,54)(23,56)(24,58)(25,60)(27,62)(28,63)(29,64)(31,65)(32,66)(33,38)(34,39)(37,55)(41,68)(43,51)(44,67)(45,70)(47,52)(48,71)(49,72)(57,69)(59,61)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6491531);