#412694

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 319890);