#489245

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,24,20)(19,28,32,47)(21,26,23,33)(22,36,39,45)(25,41,29,31)(27,35,34,38)(30,46,40,44)(37,42,48,43);(4,13)(7,12)(9,16)(11,14)(17,19,24,32)(18,25,20,29)(21,30,23,40)(22,37,39,48)(26,42,33,43)(27,44,34,46)(28,45,47,36)(31,35,41,38);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,20,24,18)(19,29,32,25)(21,26,23,33)(22,27,39,34)(28,41,47,31)(30,42,40,43)(35,45,38,36)(37,44,48,46);(17,21,24,23)(18,26,20,33)(19,30,32,40)(22,35,39,38)(25,42,29,43)(27,36,34,45)(28,46,47,44)(31,37,41,48)

Polycyclic presentation:

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

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, 396441);