#506390

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413586);