#4419620

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=h,e^2=f,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, 4326816);