#489153

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	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 686 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) = 128

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 396349);