Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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)(6,10)(7,12)(13,14,18,16)(15,21,22,19)(17,20,26,24)(23,28,25,27)(30,31)(32,39)(33,41)(42,43);(5,6)(7,11)(8,12)(9,10)(14,16)(15,22)(20,24)(23,25)(29,30,34,31)(32,37,39,35)(33,36,41,38)(40,43,44,42);(6,10)(8,11)(13,15,18,22)(14,19,16,21)(17,25,26,23)(20,28,24,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16,18,14)(15,21,22,19)(17,24,26,20)(23,28,25,27)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,8,9,11)(6,12,10,7)(13,17,18,26)(14,20,16,24)(15,23,22,25)(19,27,21,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=g,d^2=h,e^2=f,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, 4508128);