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 | |
4300 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,22,23)(19,20,27,24)(21,29,31,25)(26,32,30,28)(34,39)(35,41)(37,45)(44,47);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(18,23)(20,24)(25,29)(28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,35)(34,36)(37,43)(38,41)(39,40)(42,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,21,22,31)(18,25,23,29)(19,26,27,30)(20,28,24,32)(33,37,38,45)(34,42,39,46)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319281);