#464718

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16509
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371914);