Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1938 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)(15,16,20,24)(17,21,26,27)(22,23,28,25)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(6,9)(10,12)(14,19)(15,20)(21,27)(22,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2)(3,4)(13,16)(14,20)(15,19)(17,25)(18,24)(21,22)(23,26)(27,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,22)(16,23)(18,26)(19,27)(20,28)(24,25)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517768);