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 | |
4168 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)(18,23)(19,26)(25,32)(27,28)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,24,26,21)(20,25,30,32)(27,29,28,31)(36,44)(40,47)(42,48)(45,46);(2,6)(7,9)(8,12)(14,15)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,23)(22,26)(25,31)(27,30)(29,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,28,32,27)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=hi,e^2=g,f^2=gi,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, 10103307);