#508311

order	512
nilpotency class	4
isoclinism	isoclinic class of #508310
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 679 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415507);