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 | #35272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30);(1,2,5,3)(4,6,8,7)(10,13)(14,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,32,31);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32);(1,3,5,2)(4,7,8,6);(1,4,5,8)(2,6,3,7)(17,21,22,29)(18,24,25,32)(19,27,20,28)(23,30,26,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284382, #72284415, #72284446, … |
| maximal normal subgroup of | #72284382, #72284415, #72284446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103897);