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 | |
4223 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)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,38,43)(35,41,45,39)(36,44,46,40)(37,42,48,47);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35)(34,39)(36,42)(37,44)(38,45)(40,48)(41,43)(46,47);(2,9)(3,4)(6,15)(13,16)(18,20)(21,29)(23,24)(26,32)(36,46)(37,48)(40,44)(42,47);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,40)(35,42)(37,41)(38,46)(39,48)(43,44)(45,47);(1,5)(2,4)(3,9)(6,13)(7,8)(10,12)(11,14)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,41)(36,47)(37,44)(38,45)(39,43)(40,48)(42,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37,38,48)(34,42,43,47)(35,44,45,40)(36,41,46,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812353);