#135265

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,33,39,20,24,31,37,32,38,44,47)(19,27,29,40,30,41,43,48,28,23,34,35,42,36,45,46);(2,5)(6,8)(10,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,35)(26,36)(31,34)(32,42)(33,43)(37,40)(38,41)(39,46)(44,45)(47,48);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15)(18,24)(19,28)(21,31)(26,38)(30,42)(33,44)(35,40)(46,48)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,e^c=egi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42461);