#137931

order	512
nilpotency class	3
isoclinism	isoclinic class of #9263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	#525	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,22,27,23,28,42,46,24,29,43,47,21,26,31,32)(19,30,34,48,33,38,45,41,35,40,25,44,39,20,36,37);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,31,45,21,39,43,34,24,35,42,36,23,33,22,25)(18,20,32,48,26,40,47,37,29,38,46,44,28,30,27,41);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,25)(19,32)(21,40)(22,41)(23,30)(24,38)(26,45)(27,33)(28,36)(29,34)(31,48)(35,46)(37,43)(39,47)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45127);