#146254

order	512
nilpotency class	4
isoclinism	isoclinic class of #146230
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=dhi,b^2=de,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,d^c=dg,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

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, 53450);