#135087

order	512
nilpotency class	4
isoclinism	isoclinic class of #134965
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	Q₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 250 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,e^2=hi,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=ehi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42283);