#4433661

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=gi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4340857);