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 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,8)(9,14)(10,15)(17,18,23,24)(19,27,28,22)(20,29,32,25)(21,31,30,26)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,2,6,7)(3,4,8,12)(5,9,13,14)(10,11,15,16)(17,19,23,28)(18,22,24,27)(20,30,32,21)(25,26,29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(5,13)(9,14)(10,15)(11,16)(18,24)(22,27)(25,29)(26,31)(34,38)(39,41)(40,44)(47,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,15)(14,16)(17,20,23,32)(18,25,24,29)(19,21,28,30)(22,31,27,26)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,21)(18,26)(19,20)(22,29)(23,30)(24,31)(25,27)(28,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,22,23,27)(18,19,24,28)(20,31,32,26)(21,29,30,25)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231502);