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 | |
4071 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,20)(21,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,34,38,41)(35,39,43,37)(36,40,46,48)(42,47,44,45);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,6)(3,4)(5,13)(15,16)(18,24)(20,23)(21,30)(27,32)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,45,48,47);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,35)(34,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,30)(31,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,48,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gi,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, 8935017);