Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1742 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)(7,11)(10,12)(13,14,18,21)(15,22,19,16)(17,25,26,20)(23,24,27,28)(29,30,34,37)(31,32,39,35)(33,41,43,36)(38,42,44,40);(6,9)(10,12)(15,19)(16,22)(23,27)(24,28)(29,31,34,39)(30,35,37,32)(33,38,43,44)(36,42,41,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,41)(43,44);(1,2)(3,4)(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28)(29,32,34,35)(30,31,37,39)(33,42,43,40)(36,44,41,38);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,21,25)(15,23,19,27)(16,24,22,28)(29,33,34,43)(30,36,37,41)(31,38,39,44)(32,40,35,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769018);