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 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,10,8,11)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,26,28,27)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(2,4)(5,7,9,8)(6,10,12,11)(14,15)(17,25)(19,21)(23,28)(31,38)(32,35)(39,44)(40,43);(1,2,3,4)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(5,6)(7,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(5,8,9,7)(6,11,12,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=gh,d^2=fg,e^2=gi,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, 1785139);