Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30,34,35)(31,33,37,43)(32,39,40,36)(38,41,44,42);(2,4)(5,6,9,7)(8,10,12,11)(14,15)(19,21)(20,22)(27,28)(30,35)(31,37)(32,40)(41,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,14)(16,21,24,19)(17,20,26,22)(23,27,25,28)(29,31)(30,33)(32,38)(34,37)(35,43)(36,42)(39,41)(40,44);(5,8,9,12)(6,10,7,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,40)(30,36,35,39)(31,38,37,44)(33,42,43,41);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33)(30,37)(31,35)(32,41)(34,43)(36,38)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=fgi,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3486943);