#244380

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
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×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151576);