#511636

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 592 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,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, 418832);