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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,30,34,35)(31,39,40,36)(32,37,42,43)(33,41,44,38);(1,2,3,4)(6,9)(10,12)(14,18)(15,21)(20,23)(22,24)(30,35)(31,40)(37,43)(38,41);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,26)(22,28)(24,27)(29,31,34,40)(30,36,35,39)(32,41,42,38)(33,43,44,37);(5,7,8,11)(6,10,9,12)(13,15,17,21)(14,19,18,26)(16,24,25,22)(20,28,23,27)(29,32)(30,37)(31,41)(33,36)(34,42)(35,43)(38,40)(39,44);(13,16,17,25)(14,20,18,23)(15,22,21,24)(19,27,26,28)(29,33)(30,38)(31,37)(32,39)(34,44)(35,41)(36,42)(40,43)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=gi,d^2=fgi,e^2=fg,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, 3471635);