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 | |
4202 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,20,26)(21,29,30,27)(25,32,28,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(3,10)(6,7)(11,16)(14,15)(18,20)(24,26)(25,28)(31,32)(33,36,38,45)(34,40,42,48)(35,44,46,37)(39,41,43,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36)(34,40)(35,44)(37,46)(38,45)(39,41)(42,48)(43,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37,38,44)(34,41,42,47)(35,45,46,36)(39,40,43,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22,23,19)(18,26,20,24)(21,27,30,29)(25,31,28,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,44)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9076562);