Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1697 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)(6,9)(10,11)(15,21)(16,23)(19,24)(20,25)(29,30,34,35)(31,36,38,43)(32,33,42,37)(39,44,41,40);(5,6,8,9)(7,11,12,10)(14,18)(19,24)(20,25)(26,28)(30,35)(31,38)(33,37)(39,41);(5,6,8,9)(7,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,36)(32,41)(33,40)(34,38)(35,43)(37,44)(39,42);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32)(30,37)(31,39)(33,35)(34,42)(36,40)(38,41)(43,44);(5,7,8,12)(6,10,9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,37)(30,32,35,42)(31,40,38,44)(36,41,43,39)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4410237);