#134845

order	512
nilpotency class	4
isoclinism	isoclinic class of #134769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42041);