#6912858

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,36,38,32)(28,37,35,31)(29,39,40,33);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,38)(26,32,34,36)(28,33,35,39)(29,37,40,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,31,26,30,35,37,34)(27,33,29,32,38,39,40,36);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,29)(26,33)(27,37)(28,36)(30,40)(31,38)(32,35)(34,39);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(29,40)(32,36)(33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6820054);