#497834

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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)(5,6,10,16,11,15,26,32,12,17,25,14,9,27,8,23)(7,13,20,28,21,29,18,33,22,30,34,31,19,35,24,36);(1,2)(3,4)(5,7)(6,13)(8,24)(9,19)(10,20)(11,21)(12,22)(14,31)(15,29)(16,28)(17,30)(18,26)(23,36)(25,34)(27,35)(32,33);(5,8,9,25,12,26,11,10)(6,14,15,23,17,16,27,32)(7,18,19,20,22,24,21,34)(13,28,29,33,30,31,35,36);(2,4)

Polycyclic presentation:

2^9|a^2=g,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405030);