#266662

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,23,20,22)(25,26)(27,32)(28,30)(29,33)(31,38)(34,39)(35,40)(36,37);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,22,17,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,40)(34,38);(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(25,28,30,26)(27,34,36,40)(29,38,31,33)(32,39,37,35);(1,4,5,3)(2,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,19)(15,22,24,23)(25,29,30,31)(26,33,28,38)(27,35,36,39)(32,40,37,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173858);