#305475

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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

630 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)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23)(27,35)(31,39)(36,40)(37,38);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17)(25,26,29,33)(27,36,35,40)(28,34,30,32)(31,38,39,37);(1,3,4,8)(2,5,6,7)(10,15)(14,23)(16,20)(17,24)(25,27,29,35)(26,31,33,39)(28,37,30,38)(32,36,34,40);(10,16)(12,22)(14,24)(15,20)(17,23)(18,21)(25,28,29,30)(26,32,33,34)(27,37,35,38)(31,36,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212671);