#489155

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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

652 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 156 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)(18,24)(19,29)(20,21)(22,34)(25,41)(26,40)(27,28)(30,31)(32,48)(33,37)(35,36)(38,47)(39,44)(42,45);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,23,26)(19,25,32,42)(20,33,35,27)(21,24,36,40)(22,37,34,28)(29,43,48,46)(30,39,47,44)(31,41,38,45);(17,19,23,32)(18,25,26,42)(20,29,35,48)(21,30,36,47)(22,38,34,31)(24,39,40,44)(27,46,33,43)(28,41,37,45);(18,26)(24,40)(25,42)(27,33)(28,37)(39,44)(41,45)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396351);