#494836

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×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×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

794 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)(5,6,11,17,10,15,8,26,12,18,27,14,9,16,20,33)(7,19,25,35,24,30,21,29,22,31,36,34,23,13,28,32);(5,7,20,28,9,23,27,36,12,22,8,21,10,24,11,25)(6,13,26,35,16,31,17,32,18,30,33,34,15,19,14,29);(2,4)(5,8,9,11,12,20,10,27)(6,14,15,33,18,17,16,26)(7,21,23,25,22,28,24,36)(13,29,19,34,30,32,31,35);(7,22)(13,30)(19,31)(21,28)(23,24)(25,36)(29,32)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 402032);