#123890

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1586 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)(9,10,14,19,15,18,13,27)(11,16,22,31,21,32,28,30)(12,23,25,33,26,37,24,17)(20,34,36,29,35,39,38,40);(2,4)(5,6)(7,8)(9,11)(10,16,18,32)(12,20)(13,22)(14,28)(15,21)(17,29,33,40)(19,30,27,31)(23,34,37,39)(24,36)(25,38)(26,35);(6,8)(9,12,13,24,15,26,14,25)(10,17,19,23,18,33,27,37)(11,20,22,36,21,35,28,38)(16,29,30,34,32,40,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,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, 31086);