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 |
| prime radical | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3977 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,26)(19,21,28,32)(20,25,22,24)(27,30,29,31)(33,34)(35,44)(36,37)(38,42)(39,43)(40,45)(41,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,29)(22,27)(23,28)(24,30)(25,31)(26,32)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(3,10)(6,7)(11,16)(14,15)(17,20,23,22)(18,24,26,25)(19,27,28,29)(21,31,32,30)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,23,32)(18,19,26,28)(20,30,22,31)(24,29,25,27)(33,36,39,41)(34,37,43,47)(35,38,45,46)(40,48,44,42);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22,23,20)(18,25,26,24)(19,27,28,29)(21,31,32,30)(33,37)(34,41)(35,42)(36,43)(38,40)(39,47)(44,46)(45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,21)(20,27)(22,29)(23,28)(24,31)(25,30)(26,32)(33,38)(34,42)(35,36)(37,44)(39,46)(40,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^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, 8635695);