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 | |
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:
(5,6,9,8)(7,11,12,10)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(2,4)(5,6,9,8)(7,10,12,11)(14,15)(17,25)(19,21)(23,28)(31,35)(32,38)(39,43)(40,44);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(5,7,9,12)(6,10,8,11)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(5,8,9,6)(7,11,12,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=i,d^2=gi,e^2=fgi,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, 1724324);