#513118

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
prime radical	#2215	C₂²
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

391 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 118 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)(12,16)(13,29)(14,34)(15,36)(17,23)(18,32)(19,31)(20,28)(22,47)(24,54)(25,56)(26,48)(27,59)(30,45)(33,66)(35,43)(37,49)(38,50)(39,52)(40,55)(41,64)(42,63)(44,71)(46,53)(51,72)(57,70)(58,62)(60,69)(61,65)(67,68);(1,3,4,8)(2,5,6,7)(10,16)(17,38)(18,39)(19,40)(20,36)(21,47)(29,48)(34,59)(41,68)(42,46)(43,69)(44,61)(45,70)(49,56)(54,72)(62,66);(3,8)(5,7)(9,11,12,22)(10,17,16,38)(13,30,26,57)(14,35,27,60)(15,37,28,25)(18,43,39,69)(19,45,40,70)(20,46,36,42)(21,48,47,29)(23,52,50,32)(24,55,51,31)(33,67,58,64)(34,41,59,68)(44,72,61,54)(49,62,56,66)(53,65,63,71);(2,6)(5,7)(10,16)(17,38)(18,39)(19,40)(20,36)(21,47)(29,48)(34,59)(41,68)(42,46)(43,69)(44,61)(45,70)(49,56)(54,72)(62,66)

Polycyclic presentation:

2^9|b^2=ei,c^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, 420314);