Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,17)(16,24)(18,19)(20,23)(21,25)(22,27)(26,28)(30,32)(33,41)(35,37)(38,44);(7,11)(10,12)(14,19)(16,25)(17,20)(22,26)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,7,9,11)(6,10,8,12)(13,15)(14,20)(16,26)(17,19)(18,23)(21,28)(22,25)(24,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,16)(14,21)(15,22)(17,28)(18,25)(19,24)(20,27)(23,26)(29,32,34,30)(31,35,39,37)(33,36,41,40)(38,43,44,42);(5,8,9,6)(7,12,11,10)(13,17,18,20)(14,15,19,23)(16,27,25,28)(21,26,24,22)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3050404);