#6461505

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

1956 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)(3,6)(4,5)(7,8)(17,18,23,28)(19,29,30,44)(20,25,35,42)(21,27,22,26)(24,37,38,32)(31,45,33,48)(34,43,36,41)(39,46,40,47);(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,32)(22,37)(23,30)(25,40)(26,29)(27,44)(28,38)(31,35)(34,47)(36,46)(39,42)(41,45)(43,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,30)(24,38)(29,44)(31,33)(32,37)(39,40)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,25)(19,31)(21,36)(22,34)(23,35)(24,39)(26,43)(27,41)(28,42)(29,45)(30,33)(32,47)(37,46)(38,40)(44,48);(17,21,23,22)(18,26,28,27)(19,32,30,37)(20,34,35,36)(24,29,38,44)(25,41,42,43)(31,46,33,47)(39,48,40,45)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6368701);