#318920

order	512
nilpotency class	3
isoclinism	isoclinic class of #262787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,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,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226116);