#489158

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	C₄×C₈	C₂⁴
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

648 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,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, 396354);