#495090

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	C₂²×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 402286);