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 | #41812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,10,6)(4,15,14,8)(5,9,13,16)(17,18,24,26)(19,27,23,25)(20,22,21,31)(28,32,29,30)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(1,3,7,10)(2,6,11,12)(4,13,14,5)(8,16,15,9)(17,19,24,23)(18,25,26,27)(20,28,21,29)(22,30,31,32)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(1,4,7,14)(2,8,11,15)(3,13,10,5)(6,16,12,9)(17,20,24,21)(18,22,26,31)(19,28,23,29)(25,30,27,32)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,4)(2,8)(3,13)(5,10)(6,16)(7,14)(9,12)(11,15)(17,21,24,20)(18,22,26,31)(19,29,23,28)(25,30,27,32)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,5,7,13)(2,9,11,16)(3,14,10,4)(6,15,12,8)(17,22)(18,21)(19,30)(20,26)(23,32)(24,31)(25,29)(27,28)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,6)(2,10)(3,11)(4,16)(5,8)(7,12)(9,14)(13,15)(17,23,24,19)(18,25,26,27)(20,29,21,28)(22,30,31,32)(33,38,39,44)(34,43,37,48)(35,36,45,46)(40,41,47,42)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050627);