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 | #52970 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(2,9)(4,14)(8,12)(11,13)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657457);