#509854

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

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

normal subgroups only

This group has 800 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:

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

Polycyclic presentation:

2^9|a^2=efi,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417050);