#6584317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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	#60462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1939 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)(9,10,14,20)(11,23,27,16)(12,18,32,47)(13,19,37,49)(15,22,30,21)(17,34,45,33)(24,43,56,65)(25,53,58,40)(26,54,59,41)(28,42,29,51)(31,46,61,69)(35,63,60,48)(36,44,62,68)(38,52,39,50)(55,67,71,66)(57,70,72,64);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,16,20,23)(12,25,32,58)(13,35,37,60)(15,28,30,29)(17,43,45,65)(18,40,47,53)(19,48,49,63)(21,42,22,51)(24,33,56,34)(26,38,59,39)(31,55,61,71)(36,57,62,72)(41,52,54,50)(44,64,68,70)(46,66,69,67);(9,12,30,34,14,32,15,33)(10,17,22,47,20,45,21,18)(11,24,29,58,27,56,28,25)(13,31,39,62,37,61,38,36)(16,40,51,43,23,53,42,65)(19,44,52,69,49,68,50,46)(26,55,60,72,59,71,35,57)(41,64,48,67,54,70,63,66);(1,4,5,3)(2,6,7,8)(9,12,14,32)(10,18,20,47)(11,25,27,58)(13,36,37,62)(15,33,30,34)(16,40,23,53)(17,21,45,22)(19,44,49,68)(24,28,56,29)(26,55,59,71)(31,38,61,39)(35,57,60,72)(41,66,54,67)(42,65,51,43)(46,52,69,50)(48,64,63,70);(9,13)(10,19)(11,26)(12,31)(14,37)(15,38)(16,41)(17,44)(18,46)(20,49)(21,50)(22,52)(23,54)(24,55)(25,57)(27,59)(28,35)(29,60)(30,39)(32,61)(33,36)(34,62)(40,64)(42,63)(43,67)(45,68)(47,69)(48,51)(53,70)(56,71)(58,72)(65,66)

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,e^b=eh,h^b=hi,d^c=dh,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, 6491513);