#493100

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400296);