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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1917 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)(15,23)(16,20)(17,31)(18,21)(19,26)(22,33)(24,41)(25,37)(27,42)(28,35)(29,39)(30,40)(32,38)(34,44)(36,43);(5,6)(7,11)(8,10)(9,12)(13,15,17,26,18,27,33,43)(14,19,22,23,21,36,31,42)(16,28,30,25,29,24,32,44)(20,37,38,35,39,34,40,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,29)(14,20,21,39)(15,24,27,28)(17,30,33,32)(19,34,36,37)(22,38,31,40)(23,41,42,35)(25,26,44,43);(7,12)(9,11)(13,16)(14,20)(15,25)(17,32)(18,29)(19,35)(21,39)(22,40)(23,37)(24,43)(26,28)(27,44)(30,33)(31,38)(34,42)(36,41);(6,10)(9,11)(14,21)(16,29)(19,36)(22,31)(23,42)(24,28)(25,44)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^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, 6910775);