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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,12,15)(10,13,16,14)(19,25)(23,26)(27,31)(30,32)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3,6,11)(2,5,9,7)(4,10,12,16)(8,14,15,13)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(34,40)(35,37)(42,46)(43,45);(4,12)(8,15)(10,16)(13,14)(18,22)(23,26)(24,28)(30,32)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,4,6,12)(2,8,9,15)(3,10,11,16)(5,14,7,13)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,5,6,7)(2,3,9,11)(4,13,12,14)(8,16,15,10)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9930585);