#316809

order	512
nilpotency class	3
isoclinism	isoclinic class of #262675
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 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)(4,5)(6,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,22)(23,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,35)(37,40)(38,39);(2,5)(6,7)(9,11)(10,14)(12,19)(13,20)(15,23)(16,18)(17,22)(21,24)(25,27,35,38,29,37,28,36)(26,30,32,39,34,40,33,31);(10,15)(11,19)(13,21)(14,22)(16,17)(18,23)(25,27,29,37)(26,31,34,39)(28,38,35,36)(30,32,40,33);(1,3,4,8)(2,6,5,7)(10,16)(14,18)(15,17)(22,23)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh,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, 224005);