#6466993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 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)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,31)(21,23)(22,26)(25,33)(27,38)(28,35)(29,44)(30,34)(32,39)(36,42)(37,45)(40,43)(41,46)(47,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,32)(22,28)(24,36)(25,39)(29,45)(30,42)(31,40)(33,43)(34,38)(46,47);(9,12,13,11)(10,14,15,16)(17,19,28,42,21,30,22,27)(18,24,35,36,23,34,26,38)(20,29,43,48,33,45,32,41)(25,37,39,46,31,44,40,47);(17,20,28,43,21,33,22,32)(18,25,35,39,23,31,26,40)(19,29,42,48,30,45,27,41)(24,37,36,46,34,44,38,47);(1,3,4,7)(2,6,5,8)(20,33)(25,31)(29,45)(32,43)(37,44)(39,40)(41,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,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, 6374189);