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 | #38910 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,13)(14,16)(17,18,23,20)(19,26,25,22)(21,28,29,24)(27,30,31,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31);(2,4)(7,8)(18,20)(19,25)(21,29)(30,32);(1,4,6,2)(3,8,5,7)(17,20,23,18)(19,26,25,22)(21,28,29,24)(27,32,31,30);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,24,20,28)(19,27,25,31)(22,30,26,32);(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239255 |
| maximal normal subgroup of | #72239255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619528);