#6469041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,27)(20,25)(21,33)(22,24)(23,29)(26,35)(28,41)(30,40)(31,38)(32,45)(34,36)(37,43)(39,44)(42,48)(46,47);(9,10)(11,16)(12,14)(13,15)(17,19,22,30)(18,23,24,36)(20,28,31,42)(21,34,35,29)(25,37,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(23,36)(25,38)(26,33)(27,40)(37,47)(39,45)(41,48);(10,14)(13,16)(17,20,22,31)(18,25,24,38)(19,28,30,42)(21,32,35,44)(23,37,36,47)(26,39,33,45)(27,41,40,48)(29,43,34,46);(1,4,5,3)(2,6,7,8)(20,31)(25,38)(28,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376237);