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 | |
4110 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)(3,7)(4,5)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,30,25,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,32)(28,30)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113958);