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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1651 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)(5,6,8,10)(7,9,12,11)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28)(30,34)(31,37)(36,40)(39,43);(5,6)(7,11)(8,10)(9,12)(14,17)(19,25)(20,23)(26,27)(29,30,33,34)(31,35,37,38)(32,36,41,40)(39,42,43,44);(5,7,8,12)(6,9,10,11)(13,15,18,22)(14,19,17,25)(16,21,24,28)(20,26,23,27)(29,30)(31,38)(32,40)(33,34)(35,37)(36,41)(39,42)(43,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,21)(17,23)(18,24)(19,26)(22,28)(25,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(5,7)(6,9)(8,12)(10,11)(13,17,18,14)(15,19,22,25)(16,23,24,20)(21,26,28,27)(29,32)(30,36)(31,39)(33,41)(34,40)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fg,d^2=fi,e^2=fh,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, 5417135);