Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(30,31)(32,39)(33,41)(43,44);(2,4)(14,16)(19,21)(20,24)(27,28)(29,30,34,31)(32,35,39,37)(33,36,41,38)(40,43,42,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,43,42,44);(5,8,9,12)(6,10,7,11)(13,16,18,14)(15,21,23,19)(17,20,26,24)(22,27,25,28)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=i,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3453714);