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 | |
4143 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31)(33,35,38,45)(34,39,42,43)(36,41,46,44)(37,40,48,47);(2,7)(4,13)(6,8)(10,16)(18,20)(21,29)(23,25)(26,32)(36,46)(37,48)(40,47)(41,44);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,45)(44,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,36,38,46)(34,40,42,47)(35,44,45,41)(37,43,48,39);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,41)(35,40)(36,43)(38,48)(39,46)(42,44)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946325);