#6837240

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26)(27,35)(28,33)(29,30)(31,34)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(25,27,30,38,31,39,28,36)(26,32,33,35,34,40,29,37);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,31,30)(26,33,34,29)(27,36,39,38)(32,35,40,37);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,29,28,34,31,33,30,26)(27,37,36,40,39,35,38,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=fi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6744436);