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 | |
1712 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)(13,14)(15,20)(16,25)(17,21)(18,19)(22,23)(24,27)(26,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(6,9)(10,12)(14,19)(16,21)(20,27)(22,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,2,3,4)(13,16)(14,17)(15,22)(18,21)(19,25)(20,26)(23,27)(24,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,19)(18,25)(20,22)(24,26)(27,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=fi,e^2=h,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, 3381585);