#513214

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
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	#33908	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420410);