#489044

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,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, 396240);