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 | |
1672 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)(7,11)(10,12)(13,14)(15,17)(16,23)(18,19)(20,24)(21,27)(22,28)(25,26)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,9)(10,12)(14,19)(15,21)(16,24)(26,28)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,17,19,27)(16,25,24,22)(20,28,23,26)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2,3,4)(13,16)(14,20)(15,22)(17,26)(18,24)(19,23)(21,25)(27,28)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,15)(16,26)(18,27)(19,21)(20,22)(23,25)(24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=fgi,e^2=fh,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, 5188951);