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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(18,19)(23,25)(24,26)(30,31)(33,34)(35,36)(37,39)(38,40)(41,44)(42,43)(45,47)(46,48);(2,7)(3,12)(4,10)(14,15)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35,39,41)(34,36,37,44)(38,42,47,48)(40,43,45,46);(2,7)(4,10)(8,11)(13,16)(20,27)(23,25)(28,32)(30,31)(34,37)(35,41)(40,45)(42,48);(1,3,6,12)(2,8,7,11)(4,13,10,16)(5,15,9,14)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,35)(37,41)(38,46)(39,44)(40,48)(42,45)(43,47);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,34)(35,36,41,44)(38,45,47,40)(42,43,48,46);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,40)(35,42)(36,43)(37,45)(39,47)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495573);