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 | |
1726 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)(6,9)(10,12)(13,14)(15,19)(16,18)(17,24)(20,26)(21,28)(22,23)(25,27)(29,30,33,36)(31,34,37,41)(32,39,40,35)(38,44,43,42);(7,11)(10,12)(14,16)(19,23)(20,24)(27,28)(31,37)(34,41)(38,43)(42,44);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,23)(17,25)(18,22)(20,28)(21,26)(24,27)(29,31)(30,34)(32,38)(33,37)(35,42)(36,41)(39,44)(40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,21)(16,24)(18,26)(19,27)(22,25)(23,28)(29,32,33,40)(30,35,36,39)(31,38,37,43)(34,42,41,44)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ghi,e^2=h,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, 5191899);