#6800000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1214 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6707196);