Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,30)(24,38)(25,34)(26,35)(28,41)(29,37)(31,36)(33,39)(40,43)(42,44);(1,2)(3,4)(6,10)(11,12)(13,15)(14,20)(16,24)(17,31)(18,26)(19,25)(21,33)(22,36)(23,34)(27,38)(28,40)(29,42)(30,39)(32,35)(37,44)(41,43);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,30,19,28,17,29)(14,21,22,27,23,37,32,41)(15,24,26,39,25,40,31,42)(20,33,36,38,34,44,35,43);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(16,28)(20,34)(21,37)(26,31)(27,41)(29,30)(35,36);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,31)(16,29,28,30)(20,35,34,36)(21,27,37,41)(24,39,40,42)(33,43,44,38)Polycyclic presentation:
2^9|c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491971);