#150222

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2309 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

436 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57418);