Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1930 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)(5,6,10,8)(7,11,9,12)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27)(29,30)(31,37)(32,40)(33,39)(34,35)(36,43)(38,44)(41,42);(5,7,10,9)(6,11,8,12)(16,25)(19,24)(22,28)(26,27)(30,35)(33,37)(36,40)(42,44);(5,8,10,6)(7,11,9,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,31,34,39)(30,33,35,37)(32,41,43,38)(36,42,40,44);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28)(29,32,34,43)(30,36,35,40)(31,38,39,41)(33,44,37,42);(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)(29,33)(30,31)(32,42)(34,37)(35,39)(36,41)(38,40)(43,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4052450);