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 | |
1710 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)(13,14,17,18)(15,19,21,26)(16,20,25,24)(22,27,23,28)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(5,6,9,8)(7,11,12,10)(14,18)(15,21)(20,24)(22,23)(31,35)(32,38)(39,43)(40,44);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,23)(17,21)(18,26)(20,28)(22,25)(24,27)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,26)(22,28)(23,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,22,21,23)(19,27,26,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=fgh,e^2=gh,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, 3271512);