#135269

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10)(11,21,24,16)(12,26)(13,30)(14,36)(15,18)(17,28)(19,34)(20,33)(22,49,50,68)(23,53,31,63)(25,40,55,48)(27,58)(29,61)(32,46)(35,62)(37,45)(38,51,47,60)(39,56,44,65)(41,66)(42,59)(43,67)(52,70,57,72)(54,69,64,71);(9,11)(10,16)(12,22)(13,31)(14,25)(15,24)(17,38)(18,21)(19,44)(20,40)(23,35)(26,49)(27,50)(28,60)(29,52)(30,53)(32,64)(33,55)(34,65)(36,48)(37,57)(39,43)(41,69)(42,47)(45,72)(46,71)(51,59)(54,66)(56,67)(58,68)(61,70)(62,63);(9,12,14,37,15,27,33,29)(10,17,36,41,18,42,20,46)(11,22,25,57,24,50,55,52)(13,32,34,28,35,66,67,59)(16,38,48,69,21,47,40,71)(19,45,62,58,43,61,30,26)(23,54,56,51,31,64,65,60)(39,70,53,49,44,72,63,68)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gi

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, 42465);