Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,8)(11,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(6,10)(11,12)(16,24)(19,26)(22,28)(25,27)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(5,6)(7,11)(8,12)(9,10)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(5,8,9,7)(6,12,10,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5537677);