#123889

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	#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,h^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, 31085);