Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1730 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)(7,11)(10,12)(13,14,18,19)(15,17,21,27)(16,23,24,20)(22,28,25,26)(29,30,34,33)(31,37,39,35)(32,40,42,36)(38,43,41,44);(6,9)(10,12)(14,19)(15,21)(16,24)(26,28)(29,31,34,39)(30,35,33,37)(32,41,42,38)(36,44,40,43);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,25)(18,21)(19,27)(20,28)(22,24)(23,26)(29,31)(30,35)(32,38)(33,37)(34,39)(36,43)(40,44)(41,42);(1,2,3,4)(13,16,18,24)(14,20,19,23)(15,22,21,25)(17,26,27,28)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,44)(39,41);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,19)(16,26)(18,27)(20,25)(22,23)(24,28)(29,33,34,30)(31,35,39,37)(32,40,42,36)(38,43,41,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5187232);