#318875

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
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

520 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)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(25,26,30,35)(27,36,28,32)(29,34,31,39)(33,37,40,38);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24)(25,27,31,37,30,28,29,38)(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)(11,19)(15,17)(20,24)(25,29,30,31)(26,34,35,39)(27,37,28,38)(32,33,36,40)

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,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, 226071);