#513119

order	512
nilpotency class	5
isoclinism	isoclinic class of #513092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 396 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,27)(13,16)(14,33)(15,36)(17,24)(18,28)(19,35)(20,30)(22,49)(23,47)(25,55)(26,48)(29,60)(31,63)(32,57)(34,46)(37,45)(38,52)(39,65)(40,59)(41,54)(42,67)(43,50)(44,71)(51,69)(53,72)(56,62)(58,66)(61,68)(64,70);(1,3,4,8)(2,5,6,7)(10,16)(17,38)(18,27)(19,39)(20,40)(21,47)(33,63)(36,57)(41,43)(42,68)(44,56)(45,69)(46,70)(48,49)(55,72)(60,66);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,34)(15,37)(16,38)(18,41)(19,45)(20,46)(21,33)(24,35)(25,30)(26,28)(27,43)(29,61)(31,64)(32,51)(36,42)(39,69)(40,70)(44,72)(47,63)(48,66)(49,60)(50,62)(52,65)(53,59)(54,71)(55,56)(57,68)(58,67);(2,6)(5,7)(9,12,13,28)(10,18,16,27)(11,22,23,26)(14,29,31,58)(15,30,32,59)(17,41,38,43)(19,44,39,56)(20,36,40,57)(21,48,47,49)(24,50,52,54)(25,51,53,37)(33,66,63,60)(34,61,64,67)(35,62,65,71)(42,70,68,46)(45,72,69,55)

Polycyclic presentation:

2^9|b^2=ei,d^2=i,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=cei,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, 420315);