#512920

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
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 ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 469 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)(5,6,11,17)(7,12,18,31)(8,13,25,37)(9,26,28,55)(10,29,27,16)(14,38,40,22)(15,41,42,23)(19,30,44,56)(20,50,45,66)(21,51,46,34)(24,43,52,36)(32,61,59,47)(33,63,60,48)(35,54,39,53)(49,64,65,58)(57,68,62,67);(1,2,3,4)(5,7)(6,12)(8,19)(9,20)(10,21)(11,18)(13,30)(14,32)(15,33)(16,34)(17,31)(22,47)(23,48)(24,49)(25,44)(26,50)(27,46)(28,45)(29,51)(35,57)(36,58)(37,56)(38,61)(39,62)(40,59)(41,63)(42,60)(43,64)(52,65)(53,67)(54,68)(55,66);(6,13)(8,22)(9,23)(10,27)(12,30)(14,15)(16,43)(17,37)(19,47)(20,48)(21,46)(24,54)(25,38)(26,35)(28,41)(29,36)(31,56)(32,33)(34,64)(39,55)(40,42)(44,61)(45,63)(49,68)(50,57)(51,58)(52,53)(59,60)(62,66)(65,67);(7,18)(12,31)(19,44)(20,45)(21,46)(30,56)(32,59)(33,60)(34,51)(47,61)(48,63)(49,65)(50,66)(57,62)(58,64)(67,68)

Polycyclic presentation:

2^9|a^2=i,b^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,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, 420116);