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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(1,3,5,11)(2,6,7,9)(4,10,13,12)(8,14,15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,7)(6,9)(8,15)(14,16)(21,29)(24,31)(26,32)(28,30)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37)(34,40)(35,43)(36,41)(38,45)(39,46)(42,47)(44,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8435326);