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)(29,30,34,38)(31,32,36,35)(33,41,42,37)(39,43,44,40);(2,4)(5,6,9,7)(8,10,12,11)(14,15)(19,21)(20,22)(27,28)(31,36)(32,35)(39,44)(40,43);(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)(29,31,34,36)(30,35,38,32)(33,39,42,44)(37,43,41,40);(5,8,9,12)(6,10,7,11)(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,44,41,39);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33,34,42)(30,37,38,41)(31,39,36,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=fgi,d^2=gi,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, 3487071);