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 | |
1630 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)(14,18)(15,21)(20,25)(23,27)(29,30,34,38)(31,32,36,35)(33,37,42,44)(39,40,41,43);(6,9)(10,12)(16,24)(20,25)(23,27)(26,28)(29,31,34,36)(30,35,38,32)(33,39,42,41)(37,43,44,40);(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,35)(32,38)(33,41)(34,36)(37,40)(39,42)(43,44);(1,2)(3,4)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,41,44,39);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,41)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=g,e^2=fh,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, 5387630);