Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1898 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)(10,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28)(30,31)(32,39)(33,41)(42,43);(6,9)(10,12)(14,18)(19,25)(20,23)(26,27)(29,30,34,31)(32,37,39,35)(33,36,41,38)(40,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,21)(17,24)(18,23)(19,26)(22,28)(25,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=f,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, 4513630);