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 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4137 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(18,23)(19,25)(24,30)(26,31);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,6)(3,5)(4,12)(15,16)(19,25)(20,27)(21,29)(24,30)(34,39)(35,43)(36,46)(37,47);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,37)(34,42)(35,45)(36,40)(38,47)(39,44)(41,43)(46,48)Polycyclic presentation:
2^9|b^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8886904);