#511673

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 130 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)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,21,19)(20,27,31,24)(22,25,34,28)(23,26,35,29)(30,45,40,32)(33,44,39,41)(36,38,43,46)(37,47,48,42);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,30,31,40)(22,36,34,43)(23,26,35,29)(24,32,27,45)(25,38,28,46)(33,42,39,47)(37,41,48,44);(3,7)(6,8)(10,14)(13,16)(17,20,21,31)(18,24,19,27)(22,37,34,48)(23,39,35,33)(25,42,28,47)(26,44,29,41)(30,43,40,36)(32,46,45,38);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,19)(24,27)(25,28)(26,29)(30,40)(36,43)(41,44)(42,47)

Polycyclic presentation:

2^9|a^2=f,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, 418869);