Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1710 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)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,27,28,23)(30,34)(31,37)(36,40)(39,41);(7,11)(10,12)(14,19)(15,20)(17,26)(23,27)(29,30,33,34)(31,38,37,35)(32,40,42,36)(39,43,41,44);(5,7,9,11)(6,10,8,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,42)(41,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,16)(14,20)(15,19)(17,27)(18,24)(21,22)(23,26)(25,28)(29,31,33,37)(30,35,34,38)(32,41,42,39)(36,44,40,43);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,21,19,25)(15,22,20,28)(16,23,24,27)(29,32)(30,36)(31,39)(33,42)(34,40)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2992725);