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 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,9,11,4)(6,13,14,8)(10,12,16,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(34,40)(35,45)(37,43)(42,47);(2,5)(4,9)(6,14)(10,16)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,5,7,2)(3,9,11,4)(6,13,14,8)(10,15,16,12)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,6)(2,8)(3,10)(4,12)(5,13)(7,14)(9,15)(11,16)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957598);