Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1854 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)(7,11)(10,12)(13,14)(15,16)(17,22)(18,19)(20,21)(23,24)(25,28)(26,27)(29,30,34,37)(31,35,39,32)(33,36,42,44)(38,43,41,40);(6,9)(10,12)(14,19)(15,21)(22,27)(23,25)(29,31,34,39)(30,32,37,35)(33,41,42,38)(36,43,44,40);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,25)(18,21)(22,24)(23,26)(27,28)(29,31)(30,35)(32,37)(33,41)(34,39)(36,40)(38,42)(43,44);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,24,26,28)(22,25,27,23)(29,32,34,35)(30,31,37,39)(33,40,42,43)(36,38,44,41);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)(29,33,34,42)(30,36,37,44)(31,38,39,41)(32,40,35,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382442);