#495100

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1148 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 402296);