#484220

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60770	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1068 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391416);