#150906

order	512
nilpotency class	5
isoclinism	isoclinic class of #150443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

290 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfhi,d^b=dg,e^b=efghi,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58102);