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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(14,18)(16,23)(19,24)(22,27)(30,31)(33,41)(35,37)(39,44);(6,10)(11,12)(14,18)(15,21)(20,25)(22,27)(29,30,34,31)(32,37,40,35)(33,38,41,36)(39,42,44,43);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,30)(32,35,40,37)(33,38,41,36)(39,42,44,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(5,8,9,7)(6,12,10,11)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fh,e^2=gh,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, 1681487);