#499398

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

671 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406594);