#6834099

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=fhi,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6741295);