#134846

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,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, 42042);