#6913375

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(11,18,21,15)(12,16,22,13)(19,23,24,20)(25,26)(27,34)(28,36)(29,30)(31,32)(33,37)(35,38)(39,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,22,24)(13,23,16,20)(25,27,30,31)(26,32,29,34)(28,37,39,35)(33,36,38,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,20)(18,23)(21,24)(25,28,27,35,30,39,31,37)(26,33,32,40,29,38,34,36);(3,8)(5,7)(10,17)(12,22)(15,18)(19,24)(25,29,30,26)(27,34,31,32)(28,38,39,33)(35,36,37,40);(2,6)(5,7)(9,13,14,16)(10,12,17,22)(11,20,21,23)(15,24,18,19)(25,27,30,31)(26,34,29,32)(28,35,39,37)(33,36,38,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=gi,d^2=i,e^2=f,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, 6820571);