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 | |
4115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(2,7)(3,10)(9,14)(11,16)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(5,13)(9,14)(11,16)(12,15)(18,22)(24,25)(26,29)(31,32)(34,39)(37,40)(41,44)(45,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8091958);