Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2296 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,28)(24,39)(25,34)(26,36)(29,37)(30,35)(31,38)(33,41)(40,43)(42,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,23,36)(16,24,29,42)(17,30,18,25)(21,33,38,43)(22,35,32,34)(27,39,37,44)(28,41,31,40);(2,4)(13,16,18,28,19,29,17,31)(14,21,22,27,23,38,32,37)(15,24,25,41,26,42,30,40)(20,33,35,39,36,43,34,44);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,41)(29,42)(31,40)(32,34)(37,44)(38,43);(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350913);