#134847

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

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,c^b=ci,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, 42043);