Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,30,34,35)(31,36,37,32)(33,38,41,42)(39,43,44,40);(1,2,4,3)(5,7,8,11)(6,9,10,12)(15,21)(18,24)(22,27)(25,28)(30,35)(31,37)(33,41)(40,43);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,18,20,24)(16,22,23,27)(19,25,26,28)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,42)(41,44);(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,32,34,36)(30,37,35,31)(33,40,41,43)(38,44,42,39);(5,6)(7,9)(8,10)(11,12)(13,16,17,23)(14,19,20,26)(15,22,21,27)(18,25,24,28)(29,33,34,41)(30,38,35,42)(31,39,37,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fgi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 502429);