#6432568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,29,36)(28,35,42,46)(32,39,44,48)(41,45,43,47);(2,5)(6,8)(17,19,22,30)(18,23,26,37)(20,28,31,42)(21,27,33,29)(24,35,38,46)(25,34,40,36)(32,41,44,43)(39,45,48,47);(9,11,12,16)(10,13,14,15)(19,27)(21,33)(23,34)(25,40)(28,41)(29,30)(32,44)(35,45)(36,37)(39,48)(42,43)(46,47);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,39)(26,38)(27,41)(29,43)(30,42)(33,44)(34,45)(36,47)(37,46)(40,48);(20,31)(24,38)(28,42)(32,44)(35,46)(39,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339764);