#312806

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=i,d^2=efh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=dh,g^c=gi

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, 220002);