#512918

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	C₂×#2263	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

472 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

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

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

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=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, 420114);