Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1688 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)(7,9)(8,10)(11,12)(13,14,18,21)(15,22,20,17)(16,24,27,19)(23,26,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(7,11)(9,12)(15,20)(17,22)(23,25)(26,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,6,8,10)(7,9,11,12)(13,15,18,20)(14,17,21,22)(16,25,27,23)(19,28,24,26)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,27)(14,19,21,24)(15,23,20,25)(17,26,22,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,9,10,12)(13,17)(14,20)(15,21)(16,26)(18,22)(19,25)(23,24)(27,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5417222);