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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1927 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)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,35)(31,36,40,43)(32,41,42,37)(33,38,44,39);(1,2,3,4)(6,9)(10,12)(15,21)(18,25)(22,27)(26,28)(30,35)(32,42)(36,43)(38,39);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,31)(30,36)(32,39)(33,41)(34,40)(35,43)(37,44)(38,42);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,34,42)(30,37,35,41)(31,39,40,38)(33,36,44,43);(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,44)(30,38,35,39)(31,37,40,41)(32,43,42,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,d^2=i,e^2=fg,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, 3405388);