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 | |
4207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(5,14)(11,13)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,43)(35,40,45,48)(36,37,41,46)(38,47,44,42);(1,2,6,7)(3,8,10,15)(4,5,9,14)(11,16,13,12)(18,19)(23,25)(24,26)(30,31)(33,35,39,45)(34,40,43,48)(36,44,41,38)(37,42,46,47);(3,10)(8,15)(11,13)(12,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,39,41)(34,37,43,46)(35,38,45,44)(40,47,48,42);(1,3)(2,8)(4,13)(5,12)(6,10)(7,15)(9,11)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,37)(35,44)(38,45)(39,41)(40,42)(43,46)(47,48);(1,4,6,9)(2,5,7,14)(3,11,10,13)(8,16,15,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37)(34,41)(35,42)(36,43)(38,40)(39,46)(44,48)(45,47);(1,5)(2,9)(3,12)(4,7)(6,14)(8,11)(10,16)(13,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,44)(34,42,43,47)(35,36,45,41)(37,40,46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081482);