#6793683

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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fi,e^2=fhi,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, 6700879);