#6704002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,28)(27,31,37,34)(29,32,39,38)(33,36,40,35);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,37)(26,31,28,34)(29,35,39,36)(32,33,38,40);(2,5)(6,8)(25,26,30,28)(27,33,37,40)(29,38,39,32)(31,36,34,35);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28,30,26)(27,34,37,31)(29,38,39,32)(33,35,40,36);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,35)(29,39)(31,33)(32,38)(34,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6611198);