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 | #2635 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18,22,25)(19,24,28,23)(20,29,30,21)(26,27,32,31)(33,34)(35,41)(36,47)(37,43)(38,48)(39,44)(40,45)(42,46);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,35,40,38)(34,41,45,48)(36,39,46,43)(37,47,44,42);(3,11)(7,8)(12,16)(13,14)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,36)(34,42)(35,39)(37,41)(38,43)(40,46)(44,48)(45,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,19)(18,24)(20,26)(21,31)(22,28)(23,25)(27,29)(30,32)(33,37,40,44)(34,43,45,39)(35,42,38,47)(36,41,46,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,20,22,30)(18,21,25,29)(19,26,28,32)(23,27,24,31)(33,38,40,35)(34,41,45,48)(36,43,46,39)(37,47,44,42);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21)(18,20)(19,27)(22,29)(23,26)(24,32)(25,30)(28,31)(33,39,40,43)(34,44,45,37)(35,46,38,36)(41,42,48,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=h,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, 8640881);