#412730

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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

1086 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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, 319926);