#394733

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=di,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, 301929);