#121809

order	512
nilpotency class	3
isoclinism	isoclinic class of #120186
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

471 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29005);