#513163

order	512
nilpotency class	5
isoclinism	isoclinic class of #513098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13697	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 110 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)(9,10,12,18)(11,22,23,16)(13,17,27,40)(14,33,28,62)(15,34,29,21)(19,45,43,30)(20,47,44,31)(24,35,52,39)(25,59,53,65)(26,38,54,51)(32,48,61,42)(36,57,49,55)(37,56,50,60)(41,64,46,63)(58,68,70,67)(66,71,69,72);(1,2)(3,6)(4,5)(7,8)(9,11,12,23)(10,16,18,22)(13,24,27,52)(14,25,28,53)(15,26,29,54)(17,39,40,35)(19,36,43,49)(20,37,44,50)(21,38,34,51)(30,57,45,55)(31,56,47,60)(32,58,61,70)(33,65,62,59)(41,69,46,66)(42,68,48,67)(63,72,64,71);(2,5)(6,8)(10,17)(11,23)(13,30)(14,31)(15,29)(16,35)(18,40)(19,20)(21,48)(22,39)(24,55)(25,60)(27,45)(28,47)(32,64)(33,41)(34,42)(36,50)(37,49)(38,68)(43,44)(46,62)(51,67)(52,57)(53,56)(58,72)(59,69)(61,63)(65,66)(70,71);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,40)(19,43)(20,44)(21,34)(24,52)(25,53)(26,54)(33,62)(41,46)(42,48)(55,57)(56,60)(58,70)(71,72)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,d^b=di,f^c=fg,g^c=gh,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, 420359);