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 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,17,21)(15,19,23,18)(16,20,25,28)(22,27,24,26)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(13,15,17,23)(14,18,21,19)(16,24,25,22)(20,27,28,26)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,27)(24,25)(26,28)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,21)(19,23)(22,26)(24,27)(25,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(13,16)(14,20)(15,22)(17,25)(18,26)(19,27)(21,28)(23,24)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=hi,e^2=g,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, 1521691);