Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,18,23)(15,22,26,20)(16,28,31,44)(17,33,27,37)(19,35,30,24)(21,38,40,32)(25,42,43,39)(29,41,34,36);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,20,23,22)(16,29,31,34)(17,30,27,19)(21,39,40,42)(24,33,35,37)(25,38,43,32)(28,36,44,41);(5,8,9,6)(7,11,12,10)(13,16,30,38,18,31,19,32)(14,21,35,44,23,40,24,28)(15,25,17,34,26,43,27,29)(20,36,33,42,22,41,37,39);(7,12)(10,11)(13,17,19,15,18,27,30,26)(14,22,24,33,23,20,35,37)(16,29,32,43,31,34,38,25)(21,36,28,39,40,41,44,42);(1,3,4,2)(14,23)(16,31)(20,22)(24,35)(25,43)(29,34)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=hi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511421);