#490902

order	512
nilpotency class	4
isoclinism	isoclinic class of #490891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=di,d^c=dgi,f^c=fh,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, 398098);