Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1952 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)(7,8)(9,10)(11,12)(13,14,17,18)(15,19,23,21)(16,24,26,20)(22,28,25,27)(29,30,35,38)(31,36,32,33)(34,37,42,44)(39,43,40,41);(6,10)(7,11)(14,18)(19,21)(20,24)(27,28)(29,31,35,32)(30,36,38,33)(34,40,42,39)(37,41,44,43);(5,7,9,11)(6,8,10,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,32,35,31)(30,36,38,33)(34,40,42,39)(37,43,44,41);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,26)(25,27)(29,33,35,36)(30,32,38,31)(34,41,42,43)(37,40,44,39);(5,8)(6,7)(9,12)(10,11)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5191768);