#146106

order	512
nilpotency class	4
isoclinism	isoclinic class of #146104
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

215 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=d,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=egi,f^c=fgh,h^c=hi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53302);