#490744

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
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	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

427 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 140 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)(4,13,10,7)(6,14,15,12)(18,23)(19,28)(20,24)(21,33)(25,40)(26,37)(27,43)(29,44)(30,34)(31,47)(32,46)(35,39)(36,41)(42,45);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)(17,18,24,37,22,26,20,23)(19,29,32,36,31,25,30,42)(21,27,39,48,35,43,33,38)(28,45,34,40,47,41,46,44);(3,11)(6,15)(9,16)(12,14)(17,19,22,31)(18,25,26,29)(20,30,24,32)(21,34,35,46)(23,36,37,42)(27,44,43,40)(28,39,47,33)(38,41,48,45);(18,26)(23,37)(25,29)(27,43)(36,42)(38,48)(40,44)(41,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=eg,c^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,c^b=cg,d^b=di,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, 397940);