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 | #45868 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,20)(19,22,27,24)(21,25,30,28)(26,29,32,31)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,27,22)(21,28,30,25)(26,29,32,31)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,6)(7,10)(8,12)(14,16)(17,19,23,27)(18,24,20,22)(21,26,30,32)(25,31,28,29)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,22,27,24)(21,28,30,25)(26,29,32,31)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,19)(20,27)(21,29)(23,24)(25,26)(28,32)(30,31)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659469);