#310988

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

655 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)(4,5)(7,8)(9,10,14,22,13,16,12,17)(11,18,21,15,20,24,19,23)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(9,11,13,20)(10,15,16,23)(12,21,14,19)(17,24,22,18)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(2,5)(6,8)(10,16)(15,23)(17,22)(18,24)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,3,4,7)(2,6,5,8)(10,17)(12,14)(15,24)(16,22)(18,23)(19,21)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,g^b=gi,d^c=di,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, 218184);