Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1590 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)(9,12)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(29,30,34,33)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,6,8,10)(7,9,11,12)(14,18)(19,21)(20,24)(27,28)(32,41)(36,40)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,31,34,39)(30,35,33,37)(32,38,41,44)(36,42,40,43);(1,2,3,4)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32,34,41)(30,36,33,40)(31,38,39,44)(35,42,37,43);(5,7)(6,9)(8,11)(10,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,33,34,30)(31,37,39,35)(32,40,41,36)(38,43,44,42)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4572424);