#513213

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	C₂×D₄	D₁₆
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

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

normal subgroups only

This group has 347 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)(11,16)(12,26)(13,31)(14,35)(15,18)(17,28)(19,33)(20,24)(21,25)(22,49)(23,52)(27,57)(29,63)(30,59)(32,48)(34,58)(36,46)(37,40)(38,50)(39,54)(41,60)(42,62)(43,66)(44,61)(45,56)(47,72)(51,70)(53,68)(55,65)(64,71)(67,69);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,18,21)(12,27,30,22)(13,23,34,55)(14,24,37,56)(17,41,42,38)(19,39,44,47)(20,40,45,35)(26,57,59,49)(28,60,62,50)(29,36,64,51)(31,52,58,65)(32,66,67,53)(33,54,61,72)(43,69,68,48)(46,71,70,63);(9,12,15,30)(10,17,18,42)(11,22,25,27)(13,32,34,67)(14,36,37,51)(16,38,21,41)(19,46,44,70)(20,48,45,69)(23,53,55,66)(24,29,56,64)(26,58,59,31)(28,61,62,33)(35,43,40,68)(39,63,47,71)(49,65,57,52)(50,72,60,54);(3,8)(5,7)(10,18)(16,21)(17,42)(19,44)(20,45)(26,59)(31,58)(35,40)(38,41)(39,47)(43,68)(46,70)(48,69)(49,57)(52,65)(63,71)

Polycyclic presentation:

2^9|a^2=e,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, 420409);