Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(9,12)(14,18)(16,23)(19,24)(22,27)(29,30)(31,37)(32,39)(33,41)(34,35)(36,43)(38,40)(42,44);(5,6,8,10)(7,9,11,12)(15,21)(19,24)(22,27)(26,28)(30,35)(32,37)(36,41)(40,42);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,32)(33,38)(34,39)(35,37)(36,40)(41,42)(43,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32)(30,31)(33,42)(34,37)(35,39)(36,44)(38,41)(40,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,43)(30,36,35,41)(31,38,39,44)(32,40,37,42)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2517958);