#512098

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 164 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=ghi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419294);