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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1952 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(7,11)(9,12)(14,18)(16,25)(19,21)(22,26)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,26)(17,23)(18,21)(20,28)(22,25)(24,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,25)(26,28)(29,33,34,41)(30,37,32,40)(31,38,39,44)(35,42,36,43)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,d^2=fghi,e^2=fg,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, 1620262);