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 | |
1644 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)(7,11)(8,9)(10,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,12)(6,10,8,11)(14,18)(15,22)(20,24)(23,28)(31,38)(33,41)(35,37)(36,40);(5,7,9,12)(6,11,8,10)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28)(29,31)(30,35)(32,37)(33,39)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fgh,d^2=g,e^2=f,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, 5122241);