#135088

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,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, 42284);