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:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(29,30,34,38)(31,39,41,35)(32,40,43,36)(33,42,44,37);(7,11)(10,12)(14,19)(15,22)(17,25)(24,27)(29,31,34,41)(30,35,38,39)(32,42,43,37)(33,36,44,40);(5,7)(6,10)(8,12)(9,11)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,31)(30,35)(32,42)(33,40)(34,41)(36,44)(37,43)(38,39);(1,2,3,4)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,32,34,43)(30,36,38,40)(31,37,41,42)(33,39,44,35);(5,8,9,6)(7,12,11,10)(13,17,18,25)(14,21,19,26)(15,23,22,28)(16,24,20,27)(29,33)(30,37)(31,40)(32,39)(34,44)(35,43)(36,41)(38,42)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=i,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, 1653643);