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 | #45046 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4011 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,39,42)(35,37,44,41)(36,45,38,40)(43,47,46,48);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(3,11)(4,13)(7,14)(8,12)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=g,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, 8657596);