#150918

order	512
nilpotency class	5
isoclinism	isoclinic class of #150425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 62 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)(9,10,12,17,14,19,27,35)(11,20,23,33,24,30,38,39)(13,18,28,34,21,16,26,32)(15,25,29,36,31,37,40,22);(1,2,5,8)(3,6,4,7)(9,11,21,36,14,24,13,22)(10,15,18,30,19,31,16,20)(12,25,28,38,27,37,26,23)(17,33,32,40,35,39,34,29);(1,3,5,4)(2,7,8,6)(11,22)(13,21)(15,20)(16,18)(23,25)(24,36)(26,28)(29,33)(30,31)(32,34)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=g,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efgh,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, 58114);