#6466603

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373799);