#6556479

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,27,22,26)(24,33,39,32)(25,35,42,36)(30,38,44,48)(43,45,47,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,27,40,28,39,26,29)(20,30,36,46,37,44,35,45)(25,38,34,47,42,48,41,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,35,23,37,22,36)(18,25,26,41,28,42,27,34)(19,30,32,45,31,44,33,46)(24,38,29,43,39,48,40,47);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^d=eh

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, 6463675);