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 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,28)(26,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(2,4)(6,9)(10,12)(14,15)(16,23)(17,25)(27,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,27,26,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,26)(18,23)(20,28)(22,27)(24,25)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(13,17)(14,20)(15,22)(16,24)(18,25)(19,27)(21,28)(23,26)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=fg,e^2=fg,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, 3014504);