#511580

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

696 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26)(19,28,31,24)(20,34,23,25)(22,27,36,43)(29,44,46,38)(30,39,33,41)(32,45,47,40)(35,48,37,42);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,31)(22,29)(23,30)(25,41)(26,28)(27,38)(32,37)(34,39)(35,47)(36,46)(40,42)(43,44)(45,48);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(19,29)(20,23)(22,37)(24,38)(25,34)(27,42)(28,44)(30,47)(31,46)(32,33)(35,36)(39,40)(41,45)(43,48);(3,8)(5,6)(11,15)(13,16)(17,20,21,23)(18,25,26,34)(19,30,31,33)(22,35,36,37)(24,39,28,41)(27,42,43,48)(29,32,46,47)(38,45,44,40)

Polycyclic presentation:

2^9|a^2=i,d^2=i,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, 418776);