Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1612 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)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,35,38)(31,36,32,33)(34,37,43,44)(39,42,40,41);(6,9)(10,12)(14,18)(16,25)(19,21)(22,26)(29,31,35,32)(30,33,38,36)(34,40,43,39)(37,42,44,41);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,32,35,31)(30,36,38,33)(34,40,43,39)(37,42,44,41);(1,2,3,4)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,33,35,36)(30,32,38,31)(34,42,43,41)(37,39,44,40);(5,7,8,11)(6,10,9,12)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,34)(30,37)(31,39)(32,40)(33,41)(35,43)(36,42)(38,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3046697);