#128561

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×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, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35757);