Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,26)(23,27)(25,28)(29,30,34,35)(31,32,38,36)(33,37,42,43)(39,40,44,41);(2,4)(5,6,9,8)(7,10,11,12)(14,18)(15,22)(20,24)(23,25)(30,35)(31,38)(37,43)(39,44);(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,41,43,40);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,26)(24,27)(29,32,34,36)(30,31,35,38)(33,41,42,40)(37,44,43,39);(5,8,9,6)(7,12,11,10)(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,33,34,42)(30,37,35,43)(31,39,38,44)(32,40,36,41)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fgh,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2698575);