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, 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:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,16,21,20)(17,22,25,27)(23,24,28,26)(30,32)(35,37)(36,40)(42,43);(7,11)(9,12)(14,19)(15,21)(17,25)(24,26)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,8,11)(6,9,10,12)(13,15,18,21)(14,20,19,16)(17,23,25,28)(22,26,27,24)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,26,25,24)(22,23,27,28)(29,32,34,30)(31,35,39,37)(33,36,41,40)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,27)(20,26)(21,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=f,e^2=fg,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, 5518686);