Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1685 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)(5,6,10,9)(7,12,8,11)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(30,32)(35,37)(36,40)(42,43);(5,6)(7,12)(8,11)(9,10)(15,22)(17,25)(19,21)(20,24)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,10,8)(6,11,9,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,30)(31,35,39,37)(33,36,41,40)(38,43,44,42);(5,9,10,6)(7,12,8,11)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544275);