Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1868 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)(6,9)(10,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,32)(33,38)(34,35)(36,37)(39,40)(41,44)(42,43);(7,11)(10,12)(15,21)(18,25)(22,27)(26,28)(30,35)(31,37)(38,43)(39,41);(5,6,8,9)(7,10,11,12)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28)(29,31)(30,36)(32,35)(33,41)(34,37)(38,40)(39,42)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,36)(30,37,35,31)(33,40,42,44)(38,41,43,39);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,42)(30,38,35,43)(31,39,37,41)(32,40,36,44)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5388916);