Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1785 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)(13,14,18,16)(15,21,22,19)(17,20,26,24)(23,28,25,27)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(5,7,9,10)(6,8,11,12)(14,16)(15,22)(20,24)(23,25)(31,35)(32,38)(39,43)(40,44);(5,8,9,12)(6,10,11,7)(13,15,18,22)(14,19,16,21)(17,25,26,23)(20,28,24,27)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(1,2)(3,4)(13,16,18,14)(15,21,22,19)(17,24,26,20)(23,28,25,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(5,6)(7,8)(9,11)(10,12)(13,17,18,26)(14,20,16,24)(15,23,22,25)(19,27,21,28)(29,33,34,42)(30,36,37,41)(31,39,35,43)(32,40,38,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4509380);