#512915

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33713	C₂
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

464 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=e,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,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420111);