#491460

order	512
nilpotency class	4
isoclinism	isoclinic class of #491437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 166 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,33)(20,26)(21,28)(22,42)(23,43)(24,31)(25,32)(27,38)(29,34)(30,41)(35,45)(36,46)(37,48)(39,44)(40,47);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,34,39,24,35,41,40,25,38,23,36,21,37,22,20)(18,26,42,48,28,46,43,27,32,47,30,45,31,44,29,33);(2,7)(6,14)(8,12)(9,15)(17,20,22,37,21,36,23,38,25,40,41,35,24,39,34,19)(18,27,29,46,31,48,30,26,32,33,43,44,28,45,42,47);(17,21,25,24)(18,28,32,31)(19,35,38,37)(20,39,40,36)(22,23,41,34)(26,44,47,46)(27,48,33,45)(29,42,43,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398656);