#341110

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 248306);