#123882

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴×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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31078);