#140811

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

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

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, 48007);