Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1838 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)(5,6,8,9)(7,11,12,10)(14,18)(15,21)(16,24)(26,28)(29,30)(31,37)(32,34)(33,36)(35,38)(39,44)(40,41)(42,43);(6,9)(7,12)(14,18)(16,24)(19,22)(23,27)(30,32)(31,38)(33,41)(43,44);(6,9)(10,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,35)(32,37)(33,42)(34,38)(36,44)(39,41)(40,43);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,44,42,43);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,41)(30,36,32,40)(31,39,38,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=gh,c^2=fg,d^2=fhi,e^2=fgh,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, 5192215);