Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,16,15,13)(17,18)(19,23)(20,24)(21,22)(28,32)(30,31);(2,4)(6,7)(9,11)(10,13)(12,16)(14,15)(18,22)(20,23)(25,26,29,27)(28,30,32,31);(2,4)(6,7)(11,15)(13,16)(17,18,21,22)(19,23,24,20)(25,27,29,26)(28,31,32,30);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,13,15,16)(17,19)(18,23)(20,22)(21,24)(25,28,29,32)(26,30,27,31);(1,4,5,2)(3,6,8,7)(9,13)(10,15)(11,12)(14,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=gi,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3937292);