#146027

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=dg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53223);