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 | |
4194 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(34,38)(36,44)(39,42)(43,45);(4,11)(8,14)(10,15)(12,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(34,38)(35,41)(40,46)(43,45);(2,7)(3,9)(8,14)(10,15)(18,25)(19,30)(20,27)(28,32)(33,34,37,38)(35,42,41,39)(36,40,44,46)(43,48,45,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,44)(46,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9952803);