Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,30,33,34)(31,37,39,35)(32,40,42,36)(38,43,41,44);(1,2,3,4)(6,9)(10,12)(14,18)(15,22)(20,24)(23,25)(30,34)(35,37)(36,40)(43,44);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,22)(23,28)(25,27)(29,30)(31,35)(32,40)(33,34)(36,42)(37,39)(38,44)(41,43);(5,7,8,11)(6,10,9,12)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,31,33,39)(30,35,34,37)(32,41,42,38)(36,44,40,43);(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,28)(22,25)(29,32,33,42)(30,36,34,40)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=fi,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3471421);