#513160

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	#33740	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)(9,10,15,21)(11,16,22,35)(12,17,29,41)(13,30,32,59)(14,33,31,20)(18,42,44,26)(19,45,46,27)(23,34,48,60)(24,54,49,70)(25,55,50,38)(28,47,56,40)(36,65,63,51)(37,67,64,52)(39,58,43,57)(53,68,69,62)(61,72,66,71);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,21,35)(12,23,29,48)(13,24,32,49)(14,25,31,50)(17,34,41,60)(18,36,44,63)(19,37,46,64)(20,38,33,55)(26,51,42,65)(27,52,45,67)(28,53,56,69)(30,54,59,70)(39,61,43,66)(40,62,47,68)(57,71,58,72);(2,5)(6,8)(10,17)(12,26)(13,27)(14,31)(16,34)(18,19)(20,47)(21,41)(23,51)(24,52)(25,50)(28,58)(29,42)(30,39)(32,45)(33,40)(35,60)(36,37)(38,68)(43,59)(44,46)(48,65)(49,67)(53,72)(54,61)(55,62)(56,57)(63,64)(66,70)(69,71);(1,3,4,7)(2,6,5,8)(11,22)(16,35)(23,48)(24,49)(25,50)(34,60)(36,63)(37,64)(38,55)(51,65)(52,67)(53,69)(54,70)(61,66)(62,68)(71,72)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,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, 420356);