Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #52129 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18,23,26)(19,20,29,24)(21,22,25,31)(27,30,32,28);(1,2,5,6)(3,7,4,8)(17,19,23,29)(18,24,26,20)(21,27,25,32)(22,28,31,30);(10,11)(14,15)(17,20,23,24)(18,19,26,29)(21,30,25,28)(22,27,31,32);(1,3,5,4)(2,7,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,30)(22,27)(23,24)(25,28)(26,29)(31,32);(9,12,13,16)(10,14,11,15)(17,21)(18,22)(19,27)(20,30)(23,25)(24,28)(26,31)(29,32);(1,4,5,3)(2,8,6,7)(17,22)(18,25)(19,28)(20,27)(21,26)(23,31)(24,32)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604498);