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 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34)(35,37)(36,41)(38,43)(39,42)(40,47)(44,48)(45,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35)(34,39)(36,44)(37,43)(38,42)(40,48)(41,46)(45,47);(2,6)(7,9)(8,12)(14,15)(21,29)(25,28)(27,30)(31,32)(33,36,38,47)(34,40,43,41)(35,44,42,45)(37,46,39,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,24,22,26)(21,28,29,25)(27,31,30,32)(33,36)(34,41)(35,45)(37,46)(38,47)(39,48)(40,43)(42,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,34)(35,37)(36,41)(38,43)(39,42)(40,47)(44,48)(45,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,23,19)(18,24,20,26)(21,27,29,30)(25,32,28,31)(33,37)(34,42)(35,43)(36,46)(38,39)(40,45)(41,44)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,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, 9943905);