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)(8,13)(10,16)(14,15)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,34)(35,41)(36,45)(37,42)(38,48)(39,40)(43,44)(46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(34,40)(37,45)(41,48)(44,47);(2,6)(7,9)(8,13)(14,15)(17,18,21,25)(19,23,27,22)(20,24,29,28)(26,31,32,30)(33,35,39,38)(34,41,40,48)(36,46,42,43)(37,44,45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,36,39,42)(34,37,40,45)(35,43,38,46)(41,47,48,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37)(34,42)(35,44)(36,40)(38,47)(39,45)(41,43)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38,39,35)(34,41,40,48)(36,46,42,43)(37,47,45,44)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017997);