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 | |
4100 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,27,23)(20,24,30,32)(26,31,28,29)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,29)(26,30)(31,32)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,6)(7,13)(8,11)(14,16)(17,19,22,27)(18,23,25,21)(20,26,30,28)(24,29,32,31)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,29)(26,30)(31,32)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,29)(22,23)(24,28)(25,27)(26,32)(30,31)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=g,f^2=g,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, 10013331);