#306376

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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

579 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)(3,6)(4,8)(5,7)(9,10)(11,18)(12,14)(13,15)(16,19)(17,21)(20,24)(22,23)(27,34)(31,39)(35,40)(36,38);(1,3,5,4)(2,6,7,8)(10,14)(12,21)(15,17)(16,18)(19,22)(23,24)(25,26)(27,35)(28,32)(29,33)(30,37)(31,36)(34,40)(38,39);(1,4,5,3)(2,6,7,8)(10,15)(14,17)(16,23)(18,24)(25,27)(26,31)(28,36)(29,34)(30,38)(32,40)(33,39)(35,37);(9,11,13,20)(10,16,15,23)(12,22,21,19)(14,18,17,24)(25,28,29,30)(26,32,33,37)(27,36,34,38)(31,40,39,35)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 213572);