#150682

order	512
nilpotency class	5
isoclinism	isoclinic class of #150564
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

342 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 54 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)(9,10,12,18,14,16,25,33)(11,20,21,35,23,32,37,40)(13,19,28,36,29,17,26,34)(15,24,31,27,30,38,39,22);(1,2)(3,6)(4,8)(5,7)(9,11,14,23)(10,15,16,30)(12,24,25,38)(13,27,29,22)(17,32,19,20)(18,35,33,40)(21,26,37,28)(31,36,39,34);(1,3,5,4)(2,6,7,8)(10,16)(12,25)(15,30)(17,19)(20,32)(21,37)(24,38)(26,28)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efgi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi

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, 57878);