#511684

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

531 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,35)(29,59)(30,46)(31,47)(32,48)(34,67)(36,68)(37,58)(38,52)(39,53)(40,54)(41,73)(42,56)(43,57)(44,51)(45,61)(49,70)(50,60)(55,72)(62,78)(63,75)(64,76)(65,80)(66,74)(69,79)(71,77);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,30)(18,24,26,46)(20,34,35,70)(22,40,38,69)(23,43,39,32)(25,49,33,67)(27,54,52,79)(28,57,53,48)(29,60,61,36)(31,64,63,42)(37,71,44,65)(41,66,72,62)(45,68,59,50)(47,76,75,56)(51,80,58,77)(55,78,73,74);(3,7)(6,8)(10,14)(13,16)(17,20,21,35)(18,25,26,33)(19,29,30,61)(22,41,38,72)(23,44,39,37)(24,45,46,59)(27,55,52,73)(28,58,53,51)(31,65,63,71)(32,66,43,62)(34,69,70,40)(36,64,60,42)(47,77,75,80)(48,78,57,74)(49,79,67,54)(50,76,68,56);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,26)(19,30)(25,33)(27,52)(28,53)(29,61)(31,63)(32,43)(34,70)(40,69)(50,68)(51,58)(55,73)(56,76)(62,66)(65,71)

Polycyclic presentation:

2^9|a^2=efi,b^2=i,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418880);