#330365

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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)(9,10,13,17,12,16,21,24)(11,18,20,23,19,14,22,15)(25,26)(27,30)(28,32)(29,35)(31,33)(34,38)(36,37)(39,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,22)(23,24)(25,27,30,26)(28,36,38,40)(29,35,31,33)(32,39,34,37);(3,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,17)(16,23)(18,24)(20,21)(25,28,30,38)(26,32,27,34)(29,39,31,37)(33,36,35,40);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(17,24)(20,22)(25,29,30,31)(26,33,27,35)(28,37,38,39)(32,40,34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237561);