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 | |
4227 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)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18,23,28)(19,29,30,24)(20,21,26,25)(22,27,31,32)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,19,23,30)(18,24,28,29)(20,27,26,32)(21,22,25,31)(35,42)(39,41)(43,44)(47,48);(2,6)(4,13)(5,12)(8,11)(20,26)(21,25)(22,31)(27,32)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20)(18,25)(19,27)(21,28)(22,29)(23,26)(24,31)(30,32)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21,23,25)(18,26,28,20)(19,22,30,31)(24,32,29,27)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,22)(18,27)(19,25)(20,29)(21,30)(23,31)(24,26)(28,32)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900275);