#137921

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	#3002	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 381 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)(17,18,22,28,23,29,42,33,24,30,43,47,21,27,32,45)(19,31,26,46,34,20,36,38,35,37,40,48,39,25,44,41);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19,32,36,21,39,43,26,24,35,42,44,23,34,22,40)(18,25,45,46,27,37,47,41,30,20,33,48,29,31,28,38);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,26)(19,33)(21,31)(22,41)(23,37)(24,25)(27,44)(28,39)(29,36)(30,40)(32,48)(34,47)(35,45)(38,43)(42,46)

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 45117);