Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1968 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)(14,18)(15,21)(16,23)(26,28)(29,30,33,37)(31,35,39,34)(32,36,41,44)(38,43,40,42);(6,9)(10,12)(15,21)(19,24)(22,27)(26,28)(29,31,33,39)(30,34,37,35)(32,40,41,38)(36,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,41)(42,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,43)(41,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,38)(33,41)(34,42)(35,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1455754);