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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1684 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)(7,11)(8,10)(9,12)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26)(29,30,34,37)(31,32,39,35)(33,36,42,44)(38,40,41,43);(5,6)(7,11)(8,10)(9,12)(17,25)(20,27)(22,28)(24,26)(29,31,34,39)(30,35,37,32)(33,41,42,38)(36,40,44,43);(5,6,8,10)(7,9,12,11)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24)(29,31)(30,32)(33,41)(34,39)(35,37)(36,43)(38,42)(40,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,19)(14,15,21,23)(17,24,25,26)(20,22,27,28)(29,32)(30,31)(33,40)(34,35)(36,38)(37,39)(41,44)(42,43);(5,7,8,12)(6,9,10,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,42)(30,36,37,44)(31,38,39,41)(32,40,35,43)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=fi,e^2=h,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, 5417053);