#491688

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
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	#19661	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 570 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=fhi,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=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398884);