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 | |
4124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34,39,40)(35,41,38,43)(36,42,46,37)(44,45,47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(34,40)(36,46)(41,43)(44,47);(2,6)(7,10)(8,12)(14,16)(18,20)(23,25)(24,28)(30,31)(33,35,39,38)(34,41,40,43)(36,44,46,47)(37,48,42,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36)(34,42)(35,44)(37,40)(38,47)(39,46)(41,45)(43,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37,39,42)(34,36,40,46)(35,45,38,48)(41,47,43,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,43,40,41)(36,47,46,44)(37,45,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=g,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, 9930331);