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 | |
4058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,9,14,15)(5,12,16,11)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,39,40)(35,41,36,37)(38,42,47,48)(43,46,44,45);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,14)(15,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(34,40)(37,41)(42,48)(45,46);(2,8)(4,14)(7,10)(11,12)(18,23)(19,26)(24,29)(27,32)(33,35,39,36)(34,41,40,37)(38,43,47,44)(42,46,48,45);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,16)(11,13)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36,39,35)(34,37,40,41)(38,44,47,43)(42,45,48,46);(1,3)(2,10)(4,12)(5,15)(6,13)(7,8)(9,16)(11,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,35)(36,40)(38,46)(39,41)(42,44)(43,48)(45,47);(1,5,6,16)(2,11,8,12)(3,9,13,15)(4,7,14,10)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,47)(34,42,40,48)(35,43,36,44)(37,45,41,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820553);