Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,35)(20,24)(21,28)(26,37)(27,40)(29,38)(31,43)(32,42)(33,41)(34,39)(36,44);(6,9)(7,11)(13,15,18,34,20,29,19,28)(14,21,35,38,24,39,23,25)(16,31,33,27,17,26,32,44)(22,40,41,43,30,36,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,26)(17,20)(18,33)(19,32)(21,36)(23,42)(24,30)(25,43)(27,28)(29,31)(34,44)(35,41)(37,38)(39,40);(6,9)(10,12)(13,17,20,16)(14,22,24,30)(15,27,29,44)(18,33,19,32)(21,37,39,43)(23,41,35,42)(25,40,38,36)(26,34,31,28);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,35)(15,28,29,34)(16,32,17,33)(21,38,39,25)(22,41,30,42)(26,44,31,27)(36,43,40,37)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056597);