#6702532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6609728);