#305768

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=fhi,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212964);