#7454383

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300884
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(25,26,29,34)(27,35,36,31)(28,33,30,32)(37,40,38,39);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,29,36)(26,31,34,35)(28,37,30,38)(32,40,33,39);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(26,32)(27,36)(28,30)(31,39)(33,34)(35,40);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,28,29,30)(26,33,34,32)(27,37,36,38)(31,39,35,40);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,34)(27,36)(32,33)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7361579);