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 | |
4154 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(33,34,38,41)(35,37,43,39)(36,40,45,47)(42,44,48,46);(3,4)(5,14)(7,10)(8,13)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30)(33,35,38,43)(34,39,41,37)(36,42,45,48)(40,46,47,44);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30)(33,36,38,45)(34,40,41,47)(35,42,43,48)(37,44,39,46);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30)(33,37)(34,35)(36,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,35)(34,37)(36,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154808);