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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,12,13,9)(11,15,14,16)(17,18,23,22)(19,24,27,28)(20,25,29,31)(21,26,30,32)(33,34)(35,38)(36,46)(37,48)(39,43)(40,45)(41,47)(42,44);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,26)(29,30)(31,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,48)(42,46)(44,47);(4,13)(9,12)(11,14)(15,16)(19,27)(20,29)(24,28)(25,31)(33,36,39,47)(34,41,43,46)(35,37,45,44)(38,42,40,48);(1,4)(2,9)(3,11)(5,14)(6,12)(7,13)(8,15)(10,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32)(33,36)(34,41)(35,44)(37,45)(38,48)(39,47)(40,42)(43,46);(1,5,7,3)(2,8,6,10)(4,14,13,11)(9,15,12,16)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,37,39,44)(34,42,43,48)(35,36,45,47)(38,41,40,46);(1,6,7,2)(3,10,5,8)(4,9,13,12)(11,15,14,16)(17,22,23,18)(19,28,27,24)(20,25,29,31)(21,26,30,32)(33,38,39,40)(34,35,43,45)(36,42,47,48)(37,46,44,41)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8629482);