Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(29,30)(31,32)(33,37)(34,35)(36,40)(38,39)(41,44)(42,43);(1,2,3,4)(14,16)(19,21)(20,24)(26,27)(30,35)(31,36)(37,43)(38,44);(1,2)(3,4)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31,34,36)(30,32,35,40)(33,38,42,44)(37,39,43,41);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,32)(30,36)(31,35)(33,41)(34,40)(37,38)(39,42)(43,44);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,37)(31,38)(32,39)(34,42)(35,43)(36,44)(40,41)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2580031);