Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,23)(15,25,28,20)(16,21,33,41)(17,22,31,39)(19,24,32,40)(26,43,44,35)(27,38,30,36)(29,42,34,37);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,30)(17,26)(18,28)(19,34)(21,38)(22,35)(23,25)(24,42)(27,33)(29,32)(31,44)(36,41)(37,40)(39,43);(1,2,3,4)(13,16)(14,21)(15,26)(17,32)(18,33)(19,31)(20,35)(22,40)(23,41)(24,39)(25,43)(27,29)(28,44)(30,34)(36,37)(38,42);(5,8,9,7)(6,10,11,12)(16,31)(17,33)(19,32)(21,39)(22,41)(24,40)(26,27)(29,34)(30,44)(35,36)(37,42)(38,43);(13,17,18,31)(14,22,23,39)(15,27,28,30)(16,32,33,19)(20,36,25,38)(21,40,41,24)(26,29,44,34)(35,37,43,42)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491577);