#6909942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(25,26,30,36)(27,35,31,32)(28,37,38,40)(29,39,33,34);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,36,35)(28,34,38,39)(29,37,33,40);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,28,27,34,30,38,31,39)(26,33,32,40,36,29,35,37);(3,8)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,33)(26,34,36,39)(27,37,31,40)(28,32,38,35);(2,6)(5,7)(11,19)(14,17)(18,24)(22,23)(25,27,30,31)(26,35,36,32)(28,34,38,39)(29,40,33,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817138);