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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,11)(7,12,10,8)(13,14)(15,19)(16,18)(17,24)(20,26)(21,28)(22,23)(25,27)(29,30,34,35)(31,36,39,44)(32,33,38,37)(40,41,42,43);(5,7,9,10)(6,8,11,12)(14,16)(19,23)(20,24)(27,28)(30,35)(31,39)(32,38)(41,43);(5,8,9,12)(6,10,11,7)(13,15)(14,19)(16,23)(17,25)(18,22)(20,28)(21,26)(24,27)(29,31)(30,36)(32,42)(33,43)(34,39)(35,44)(37,41)(38,40);(1,2,3,4)(5,7)(6,8)(9,10)(11,12)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,32,34,38)(30,37,35,33)(31,40,39,42)(36,43,44,41);(5,8)(6,7)(9,12)(10,11)(13,17)(14,20)(15,21)(16,24)(18,26)(19,27)(22,25)(23,28)(29,33,34,37)(30,38,35,32)(31,41,39,43)(36,42,44,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=gh,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, 5190427);