#149735

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator, prime radical	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,28,23,29,36,44,24,25,34,38,37,39,46,47,22,27,31,40,33,41,45,48,19,30,20,26,32,42,35,43);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,24)(18,25,27,30)(20,21,34,31)(23,32,33,37)(26,40,38,28)(29,39,41,42)(35,36,46,45)(43,48,47,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,33,46,24,21,32,45,22,34,23,35,19,31,37,36)(18,26,41,47,25,28,42,48,27,38,29,43,30,40,39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56931);