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 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,8,14,16)(4,5,10,9)(6,15,13,11)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(34,39)(35,40)(37,45)(43,46);(1,3,7,14)(2,8,12,16)(4,13,10,6)(5,11,9,15)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(34,39)(35,40)(41,47)(42,48);(1,4)(2,9)(3,13)(5,12)(6,14)(7,10)(8,15)(11,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,4)(2,5)(3,13)(6,14)(7,10)(8,11)(9,12)(15,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,5)(2,10)(3,11)(4,12)(6,8)(7,9)(13,16)(14,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,6,7,13)(2,11,12,15)(3,4,14,10)(5,8,9,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,41)(35,42)(36,43)(38,45)(39,47)(40,48)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441542);