Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| 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:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,8,11)(6,9,10,12)(14,18)(16,25)(19,21)(22,26)(33,41)(36,40)(38,44)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,26)(17,23)(18,21)(20,28)(22,25)(24,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,25)(26,28)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,28)(23,26)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5120672);