Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1832 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)(13,14,18,19)(15,16,21,24)(17,20,27,28)(22,23,25,26)(29,30,34,37)(31,38,35,32)(33,41,44,36)(39,40,42,43);(6,9)(10,12)(14,19)(15,21)(20,28)(22,25)(29,31,34,35)(30,32,37,38)(33,42,44,39)(36,43,41,40);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,25)(18,21)(19,24)(20,26)(22,27)(23,28)(29,31)(30,32)(33,42)(34,35)(36,43)(37,38)(39,44)(40,41);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,26)(18,24)(19,21)(20,25)(22,28)(23,27)(29,32)(30,35)(31,37)(33,43)(34,38)(36,39)(40,44)(41,42);(13,17,18,27)(14,20,19,28)(15,22,21,25)(16,23,24,26)(29,33)(30,36)(31,39)(32,40)(34,44)(35,42)(37,41)(38,43)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4049814);