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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,11,12)(7,8,9,10)(13,14,17,18)(15,19,21,22)(16,20,25,26)(23,27,24,28)(29,30)(31,37)(32,39)(33,41)(34,35)(36,42)(38,40)(43,44);(5,7,11,9)(6,8,12,10)(14,18)(15,21)(20,26)(23,24)(30,35)(32,37)(33,42)(38,43);(5,8,11,10)(6,7,12,9)(13,15)(14,19)(16,24)(17,21)(18,22)(20,28)(23,25)(26,27)(29,31)(30,32)(33,43)(34,39)(35,37)(36,44)(38,42)(40,41);(1,2,3,4)(5,9,11,7)(6,8,12,10)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,27)(25,26)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,43)(42,44);(5,10,11,8)(6,7,12,9)(13,16,17,25)(14,20,18,26)(15,23,21,24)(19,27,22,28)(29,33)(30,36)(31,38)(32,40)(34,42)(35,41)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fgh,d^2=fi,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, 5188186);