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 | |
1614 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)(7,9)(8,10)(11,12)(13,14,18,19)(15,20,25,27)(16,21,22,17)(23,24,28,26)(29,30,34,37)(31,38,35,32)(33,36,42,44)(39,43,41,40);(8,12)(10,11)(14,19)(17,21)(20,27)(24,26)(29,31,34,35)(30,32,37,38)(33,39,42,41)(36,40,44,43);(6,7)(10,11)(13,15,18,25)(14,20,19,27)(16,23,22,28)(17,26,21,24)(29,31)(30,32)(33,41)(34,35)(36,43)(37,38)(39,42)(40,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,22)(14,21,19,17)(15,23,25,28)(20,24,27,26)(29,32)(30,35)(31,37)(33,40)(34,38)(36,41)(39,44)(42,43);(5,8)(6,10)(7,11)(9,12)(13,17,18,21)(14,22,19,16)(15,24,25,26)(20,23,27,28)(29,33,34,42)(30,36,37,44)(31,39,35,41)(32,40,38,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fg,e^2=f,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, 5382736);