#136661

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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 104 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eg,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43857);