#6814228

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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

1180 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,30)(28,39)(29,37)(31,36)(32,35)(33,38)(34,40);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,33)(28,29)(30,36)(31,38)(32,40)(34,37)(35,39);(2,6)(5,8)(25,28,29,27,31,40,32,38)(26,34,37,33,36,39,35,30);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,32)(26,35,36,37)(27,28,38,40)(30,34,33,39);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,29,34,31,33,32,39)(26,27,37,40,36,38,35,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721424);