#141956

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,7)(4,5)(6,8)(9,10)(11,13,15,16)(12,14)(17,18,22,27,23,26,29,42,21,25,36,41,20,34,35,38)(19,28,32,44,33,39,45,48,31,43,46,47,30,24,37,40);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,19,29,45,20,30,22,32,21,31,35,37,23,33,36,46)(18,24,38,47,34,43,41,48,25,39,42,44,26,28,27,40);(1,3,4,8)(2,6,5,7)(10,14)(13,16)(18,25)(24,39)(26,34)(27,41)(28,43)(38,42)(40,48)(44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,f^b=fgi

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, 49152);