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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,13)(14,15)(18,20)(19,23)(25,26,30,28)(27,31,29,32);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(19,23)(22,24)(25,27,30,29)(26,31,28,32);(3,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,32,28,31);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,31,29,32);(1,5,6,3)(2,8,4,7)(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=fgh,d^2=ghi,e^2=gh,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, 3902381);