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 | |
4288 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)(3,7)(4,10)(5,9)(6,8)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,3,6,10)(2,7,8,4)(5,11,15,13)(9,14,16,12)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(2,8)(3,10)(9,16)(11,13)(19,26)(22,23)(27,32)(30,31)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3)(2,4)(5,13)(6,10)(7,8)(9,14)(11,15)(12,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,4)(2,3)(5,14)(6,7)(8,10)(9,13)(11,16)(12,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,5,6,15)(2,9,8,16)(3,11,10,13)(4,12,7,14)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8636006);