#128519

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,30)(24,31)(25,29)(26,34)(27,32)(28,33)(35,41)(36,39)(37,40)(38,42)(43,44);(6,9,11,10)(13,15,16,23,17,24,26,35,18,25,27,36,28,37,38,43)(14,19,21,30,22,31,34,41,20,29,32,39,33,40,42,44);(1,3,4,2)(14,20)(19,29)(21,32)(22,33)(30,39)(31,40)(34,42)(41,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg

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, 35715);