#323798

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

592 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)(9,10,13,17,12,16,21,15)(11,18,20,23,19,14,22,24)(26,31)(27,35)(33,34)(37,38);(1,3,5,4)(2,6,8,7)(9,11,12,19)(10,14,16,18)(13,20,21,22)(15,23,17,24)(25,26,28,33,29,31,30,34)(27,36,38,32,35,40,37,39);(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(25,27,29,35)(26,32,31,39)(28,37,30,38)(33,36,34,40);(1,4,5,3)(2,7,8,6)(10,16)(11,19)(15,17)(20,22)(25,28,29,30)(26,33,31,34)(27,37,35,38)(32,36,39,40)

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,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, 230994);