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 | |
3996 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)(5,13)(6,8)(7,11)(9,14)(10,15)(12,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,26)(27,28)(29,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19,23,29)(18,24,26,32)(20,25,31,27)(21,22,28,30)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,8)(4,7)(5,14)(10,16)(17,20,23,31)(18,22,26,30)(19,27,29,25)(21,32,28,24)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,20)(18,22)(19,25)(21,32)(23,31)(24,28)(26,30)(27,29)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,13,16)(17,21,23,28)(18,25,26,27)(19,22,29,30)(20,24,31,32)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,5,6,14)(2,9,8,13)(3,10,11,16)(4,12,7,15)(17,22)(18,20)(19,28)(21,29)(23,30)(24,25)(26,31)(27,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961145);