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) = C26×C4 |
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 | |
1836 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)(9,12)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(5,6,8,10)(7,9,11,12)(14,18)(19,21)(20,24)(27,28)(31,38)(32,35)(39,44)(40,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=h,e^2=f,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, 4508111);