#509850

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

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, 417046);