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 | |
3971 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(2,7)(3,8)(9,16)(10,13)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(35,41)(38,45)(42,47)(46,48);(2,7)(3,8)(9,16)(10,13)(19,28)(23,24)(29,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=gh,f^2=gi,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, 8653711);