Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1588 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28)(29,30,34,35)(31,32,37,40)(33,36,42,43)(38,39,41,44);(5,7,9,12)(6,8,10,11)(14,16)(19,21)(20,24)(27,28)(30,35)(31,37)(36,43)(38,41);(5,6,9,10)(7,11,12,8)(13,15)(14,19)(16,21)(17,25)(18,23)(20,28)(22,26)(24,27)(29,31)(30,32)(33,41)(34,37)(35,40)(36,44)(38,42)(39,43);(1,2,3,4)(5,8)(6,7)(9,11)(10,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,32)(30,31)(33,39)(34,40)(35,37)(36,38)(41,43)(42,44);(5,6)(7,8)(9,10)(11,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,33,34,42)(30,36,35,43)(31,38,37,41)(32,39,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fgh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575763);