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 | |
1630 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)(5,6,10,12)(7,9,8,11)(15,21)(19,24)(22,27)(26,28)(29,30,33,37)(31,35,39,34)(32,36,41,44)(38,43,40,42);(5,7)(6,11)(8,10)(9,12)(14,18)(19,24)(20,25)(26,28)(29,31,33,39)(30,34,37,35)(32,40,41,38)(36,43,44,42);(1,3,4,2)(6,12)(7,8)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,41)(42,44);(5,8,10,7)(6,9,12,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,43)(41,44);(5,9)(6,7)(8,12)(10,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,38)(33,41)(34,42)(35,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fi,d^2=fgh,e^2=fh,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, 499810);