Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1716 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)(7,9,11,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30,34,35)(31,38,36,32)(33,37,41,43)(39,42,44,40);(5,7,8,11)(6,9,10,12)(15,22)(17,25)(19,21)(20,24)(30,35)(31,36)(33,41)(40,42);(6,10)(9,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31,34,36)(30,32,35,38)(33,39,41,44)(37,40,43,42);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32)(30,36)(31,35)(33,42)(34,38)(37,39)(40,41)(43,44);(5,7)(6,9)(8,11)(10,12)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33)(30,37)(31,39)(32,40)(34,41)(35,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5543771);