Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,25)(26,27)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(14,19)(17,20)(21,27)(25,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,17)(16,24,26,23)(21,25,27,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,28)(22,24)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(1,3,4,2)(13,17,18,20)(14,22,19,15)(16,25,26,28)(21,24,27,23)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgi,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 507009);