#493690

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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×C₄⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1212 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)(5,6,10,16,11,15,21,31,12,17,20,13,9,26,7,18)(8,14,24,30,25,29,33,36,22,28,34,27,23,35,19,32);(5,7,9,20,12,21,11,10)(6,13,15,18,17,16,26,31)(8,19,23,34,22,33,25,24)(14,27,29,32,28,30,35,36);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,22)(13,27)(15,29)(16,30)(17,28)(18,32)(20,34)(21,33)(26,35)(31,36);(8,22)(14,28)(19,33)(23,25)(24,34)(27,30)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^c=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, 400886);