#266556

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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

688 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)(9,10,15,11)(12,20,24,21)(13,19,14,17)(16,23,18,22)(25,26)(27,31)(28,36)(29,38)(30,34)(32,37)(33,39)(35,40);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,21,24,20)(13,19,14,17)(16,23,18,22)(25,27,30,28)(26,31,34,36)(29,37,39,35)(32,33,40,38);(9,12,15,24)(10,16,11,18)(13,23,14,22)(17,21,19,20)(25,28,30,27)(26,32,34,40)(29,35,39,37)(31,38,36,33);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=ehi,b^2=eh,c^2=h,d^2=e,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,f^c=fh,g^c=ghi

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, 173752);