#506362

order	512
nilpotency class	4
isoclinism	isoclinic class of #506362
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28225
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413558);