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 | #52565 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,6)(4,5,9,15)(12,13,16,14)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(18,24)(19,20)(26,31)(27,28)(34,40)(35,41)(42,43)(44,45);(3,11)(6,8)(12,16)(13,14)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38,39,37)(34,43,40,42)(35,45,41,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192736);