#6434310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1694 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,35)(21,27)(22,28)(23,41)(24,29)(25,33)(26,37)(31,47)(32,44)(34,42)(36,43)(38,48)(39,45)(40,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,33,24,34,22,36)(18,25,28,30,29,43,41,42)(20,31,37,40,38,21,32,39)(26,27,35,46,44,47,48,45);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,23,37,24,38,22,32)(18,26,28,35,29,44,41,48)(19,31,33,40,34,21,36,39)(25,27,30,46,43,47,42,45);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,24,31)(18,27,29,47)(19,32,34,37)(20,36,38,33)(22,40,23,39)(25,35,43,48)(26,30,44,42)(28,46,41,45);(17,22,24,23)(18,28,29,41)(19,33,34,36)(20,32,38,37)(21,39,31,40)(25,42,43,30)(26,35,44,48)(27,45,47,46)

Polycyclic presentation:

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

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, 6341506);