#490917

order	512
nilpotency class	4
isoclinism	isoclinic class of #490883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
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

478 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 130 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,28)(20,26)(21,24)(22,35)(23,38)(25,31)(27,37)(29,43)(30,44)(32,48)(33,40)(34,45)(36,42)(39,47)(41,46);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(18,24)(25,39)(26,38)(27,40)(28,44)(35,45)(41,42)(43,48);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19,21,30)(18,25,24,39)(20,32,23,29)(22,36,34,46)(26,42,38,41)(27,43,40,48)(28,45,44,35)(31,33,47,37);(3,10)(8,15)(11,14)(12,16)(17,20,21,23)(18,26,24,38)(19,29,30,32)(22,33,34,37)(25,41,39,42)(27,35,40,45)(28,43,44,48)(31,46,47,36)

Polycyclic presentation:

2^9|b^2=egi,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^c=dgi,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398113);