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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(2,4)(5,6,9,7)(8,10,12,11)(14,18)(16,24)(19,21)(22,28)(30,34)(32,41)(35,37)(38,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,39)(30,35,34,37)(32,38,41,44)(36,42,40,43);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=gi,e^2=f,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, 1720218);