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 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(19,20)(21,29)(23,26)(24,31)(33,34)(35,42)(36,41)(37,47)(38,40)(39,43)(44,45)(46,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,35,39,38)(34,40,43,42)(36,44,47,46)(37,45,41,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36,39,47)(34,41,43,37)(35,44,38,46)(40,48,42,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,37)(34,36)(35,45)(38,48)(39,41)(40,44)(42,46)(43,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,34)(35,40)(36,41)(37,47)(38,42)(39,43)(44,48)(45,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,38,39,35)(34,42,43,40)(36,46,47,44)(37,48,41,45)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044563);