Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,18,15)(16,21,23,19)(17,20,26,22)(24,28,25,27)(29,30)(31,34)(32,37)(33,38)(35,40)(36,42)(39,43)(41,44);(1,2,3,4)(6,9)(10,12)(14,15)(16,23)(20,22)(24,25)(30,31)(35,37)(36,38)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,20,26,22)(24,27,25,28)(29,31,34,30)(32,37,40,35)(33,36,42,38)(39,43,41,44);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,25)(18,23)(20,28)(22,27)(24,26)(29,32)(30,35)(31,37)(33,41)(34,40)(36,44)(38,43)(39,42);(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,33,34,42)(30,36,31,38)(32,39,40,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=f,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3405382);