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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(10,14)(11,16)(12,15)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34)(35,40)(36,43)(37,42)(38,41)(39,44)(45,47)(46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(33,35)(34,37)(36,45)(38,46)(39,42)(40,44)(41,47)(43,48);(2,7)(4,8)(10,14)(12,15)(18,24)(23,26)(25,28)(31,32)(33,35)(34,40)(36,45)(37,44)(38,46)(39,42)(41,48)(43,47);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,36,39,38)(34,41,44,43)(35,45,42,46)(37,47,40,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32)(33,37,39,40)(34,42,44,35)(36,47,38,48)(41,46,43,45);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21,22,19)(18,26,24,23)(20,29,30,27)(25,32,28,31)(33,38,39,36)(34,43,44,41)(35,46,42,45)(37,48,40,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8028633);