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 | #42992 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1895 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,30)(17,21)(18,34)(19,22)(20,28)(23,25)(26,36)(27,38)(29,42)(31,41)(32,39)(33,40)(35,44)(37,43);(5,6)(7,11)(8,10)(9,12)(13,15,25,43,19,29,18,28)(14,20,34,42,22,37,23,24)(16,31,33,27,17,26,32,44)(21,38,40,41,30,35,39,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,19)(18,32)(20,35)(22,30)(23,39)(24,41)(25,33)(27,28)(29,31)(34,40)(36,42)(37,38)(43,44);(7,12)(9,11)(13,17,19,16)(14,21,22,30)(15,27,29,44)(18,32,25,33)(20,36,37,41)(23,40,34,39)(24,38,42,35)(26,43,31,28);(6,10)(9,11)(14,22)(16,17)(20,37)(23,34)(24,42)(26,31)(27,44)(32,33)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910797);