Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1795 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)(5,6,8,10)(7,9,11,12)(13,14,18,17)(15,19,21,23)(16,24,26,20)(22,28,25,27)(29,30,35,38)(31,36,32,33)(34,42,43,37)(39,41,40,44);(7,11)(9,12)(14,17)(15,21)(20,24)(22,25)(29,31,35,32)(30,33,38,36)(34,40,43,39)(37,44,42,41);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,32,35,31)(30,36,38,33)(34,40,43,39)(37,44,42,41);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,24)(18,26)(19,27)(21,25)(23,28)(29,33,35,36)(30,32,38,31)(34,41,43,44)(37,40,42,39);(13,17,18,14)(15,23,21,19)(16,24,26,20)(22,28,25,27)(29,34,35,43)(30,37,38,42)(31,39,32,40)(33,41,36,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1620097);