#6432681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fg,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, 6339877);