Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1706 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)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,6,8,10)(7,9,11,12)(14,18)(16,25)(19,21)(22,28)(32,41)(35,40)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2)(3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44);(5,7,8,11)(6,9,10,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1513913);