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 | |
1608 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)(5,6,9,8)(7,10,12,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,34,35)(31,33,37,43)(32,39,42,36)(38,44,40,41);(5,7,9,12)(6,10,8,11)(14,17)(16,25)(19,21)(22,26)(30,35)(33,43)(36,39)(41,44);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,26)(17,21)(18,23)(20,28)(22,25)(24,27)(29,31,34,37)(30,33,35,43)(32,40,42,38)(36,44,39,41);(1,2,3,4)(13,16)(14,20)(15,22)(17,24)(18,25)(19,27)(21,28)(23,26)(29,32,34,42)(30,36,35,39)(31,38,37,40)(33,41,43,44);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,21,23,19)(16,24,25,20)(22,28,26,27)(29,33)(30,37)(31,35)(32,41)(34,43)(36,40)(38,39)(42,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=gi,e^2=gh,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, 1653947);