#512921

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	#33720	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 467 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)(7,17)(8,21)(9,23)(10,27)(12,30)(14,37)(15,39)(16,36)(18,46)(19,49)(20,45)(22,26)(24,53)(25,43)(28,47)(29,44)(31,57)(32,60)(33,62)(34,56)(35,61)(38,41)(40,67)(42,58)(48,54)(50,51)(52,55)(59,68)(63,65)(64,66);(1,2,3,4)(5,6)(7,12)(8,14)(9,15)(10,16)(11,13)(17,30)(18,32)(19,33)(20,31)(21,37)(22,38)(23,39)(24,40)(25,34)(26,41)(27,36)(28,35)(29,42)(43,56)(44,58)(45,57)(46,60)(47,61)(48,59)(49,62)(50,64)(51,66)(52,63)(53,67)(54,68)(55,65);(5,7,11,20)(6,12,13,31)(8,22,25,52)(9,26,28,55)(10,29,27,19)(14,38,34,63)(15,41,35,65)(16,42,36,33)(17,43,45,21)(18,47,48,23)(24,54,50,46)(30,56,57,37)(32,61,59,39)(40,68,64,60)(44,51,49,53)(58,66,62,67);(6,13)(12,31)(14,34)(15,35)(16,36)(30,57)(32,59)(33,42)(37,56)(38,63)(39,61)(40,64)(41,65)(58,62)(60,68)(66,67)

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,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, 420117);