#416197

order	512
nilpotency class	3
isoclinism	isoclinic class of #395581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi

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, 323393);