#323358

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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

616 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)(9,10,12,17,13,16,20,15)(11,14,18,23,19,22,24,21)(25,26,29,35,31,34,30,36)(27,32,37,33,28,39,38,40);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,16,22)(12,18,20,24)(15,21,17,23)(25,27,29,37,31,28,30,38)(26,32,35,33,34,39,36,40);(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,28,31,27)(26,33,34,40)(29,37,30,38)(32,36,39,35);(1,4,5,3)(2,7,8,6)(10,16)(14,22)(15,17)(21,23)(26,34)(32,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230554);