Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,26)(24,25)(27,28)(30,31)(32,39)(33,41)(43,44);(7,11)(10,12)(14,19)(15,21)(22,28)(23,25)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,44,42,43);(5,7,9,11)(6,10,8,12)(13,15,18,21)(14,20,19,16)(17,25,27,23)(22,24,28,26)(29,31,34,30)(32,35,39,37)(33,36,41,38)(40,44,42,43);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,26,27,24)(22,23,28,25)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(5,8,9,6)(7,12,11,10)(13,17,18,27)(14,22,19,28)(15,23,21,25)(16,24,20,26)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=gh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4059426);