#136769

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
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	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128

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

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

Polycyclic presentation:

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

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, 43965);