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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(10,14)(11,16)(12,15)(18,20)(22,24)(25,28)(29,31)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(2,7)(4,8)(10,14)(12,15)(17,18,23,20)(19,24,27,22)(21,28,30,25)(26,29,32,31)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19,23,27)(18,24,20,22)(21,26,30,32)(25,31,28,29)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,20,23,18)(19,22,27,24)(21,28,30,25)(26,29,32,31)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,30)(18,25,20,28)(19,26,27,32)(22,29,24,31)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22)(18,19)(20,27)(21,29)(23,24)(25,26)(28,32)(30,31)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017648);