#146186

order	512
nilpotency class	4
isoclinism	isoclinic class of #146186
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
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

213 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53382);