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 | #59095 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(4,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,26,24,20)(21,25,31,32)(27,30,29,28);(1,2,6,5)(3,8,7,4)(9,11,12,15)(10,13,14,16)(18,23)(19,24)(25,32)(27,29);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,24)(25,32)(27,29);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32);(1,5,6,2)(3,8,7,4)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8629390);