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 | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,42)(35,40,45,38)(36,37,48,41)(43,47,46,44);(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(18,24)(19,20)(26,29)(27,28)(33,35,39,45)(34,40,42,38)(36,46,48,43)(37,44,41,47);(2,7)(3,9)(8,15)(10,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,39,48)(34,41,42,37)(35,43,45,46)(38,47,40,44);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36)(34,37)(35,43)(38,44)(39,48)(40,47)(41,42)(45,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37)(34,36)(35,44)(38,43)(39,41)(40,46)(42,48)(45,47);(1,5)(2,9)(3,7)(4,11)(6,14)(8,10)(12,13)(15,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38)(34,35)(36,47)(37,46)(39,40)(41,43)(42,45)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=g,e^2=g,f^2=g,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, 8616404);