#150687

order	512
nilpotency class	5
isoclinism	isoclinic class of #150584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 284 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)(11,20,23,30)(12,16,24,32)(13,18,27,17)(14,19)(15,22,29,26)(21,34,36,28)(25,33)(31,38)(35,40,37,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15)(12,21,24,35)(13,26,27,23)(16,28)(17,20)(18,29)(19,30)(25,36,38,37)(31,34)(32,39)(33,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,31)(18,33)(19,32)(20,34)(22,35)(23,37)(26,36)(27,38)(29,40)(30,39)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=gi,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=efg,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, 57883);