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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1804 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)(6,7)(10,11)(13,14,18,19)(15,20,21,16)(17,25,27,22)(23,24,26,28)(29,30)(31,33)(32,39)(34,36)(35,37)(38,41)(40,43)(42,44);(5,6)(7,9)(8,10)(11,12)(14,19)(16,20)(22,25)(24,28)(29,31,35,32)(30,33,37,39)(34,40,42,38)(36,43,44,41);(6,7)(8,12)(13,15)(14,20)(16,19)(17,26)(18,21)(22,24)(23,27)(25,28)(29,32,35,31)(30,33,37,39)(34,40,42,38)(36,41,44,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,21,19,15)(17,24,27,28)(22,26,25,23)(29,33)(30,32)(31,37)(34,41)(35,39)(36,40)(38,44)(42,43);(5,8)(6,10)(7,11)(9,12)(13,17,18,27)(14,22,19,25)(15,23,21,26)(16,24,20,28)(29,34,35,42)(30,36,37,44)(31,38,32,40)(33,41,39,43)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=g,d^2=gi,e^2=fgh,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, 5190444);