#312576

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

646 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)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24)(25,26)(27,31)(28,30)(29,32)(33,40)(34,37)(35,38)(36,39);(1,3,4,8)(2,5,6,7)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(26,30)(27,33)(29,35)(31,39)(34,36)(37,40);(2,6)(5,7)(25,27,28,34)(26,31,30,37)(29,36,35,33)(32,39,38,40);(2,6)(5,7)(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)(26,30)(31,37)(32,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219772);