#495011

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402207);