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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1724 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)(5,6,9,8)(7,10,11,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(30,34)(31,37)(32,40)(42,44);(7,11)(10,12)(14,15)(17,25)(19,21)(23,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,7,9,11)(6,10,8,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(1,2,3,4)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(5,8,9,6)(7,12,11,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=fgh,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, 1621297);