#6466994

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=f,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, 6374190);