Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,28)(26,27)(29,30,34,38)(31,36,41,35)(32,33,43,37)(39,40,42,44);(7,11)(9,12)(14,19)(16,23)(21,27)(24,28)(29,31,34,41)(30,35,38,36)(32,42,43,39)(33,40,37,44);(5,6,8,10)(7,9,11,12)(13,15)(14,16)(17,25)(18,20)(19,23)(21,28)(22,26)(24,27)(29,31)(30,36)(32,42)(33,44)(34,41)(35,38)(37,40)(39,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,25)(22,27)(26,28)(29,32)(30,37)(31,39)(33,38)(34,43)(35,40)(36,44)(41,42);(5,7)(6,9)(8,11)(10,12)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,25)(23,28)(29,33,34,37)(30,32,38,43)(31,40,41,44)(35,42,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990923);