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 | #47553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,11,8)(5,9,13,14)(10,12,16,15)(19,26)(22,24)(27,32)(29,31)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(2,7)(4,8)(5,13)(10,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(1,3,6,11)(2,4,7,8)(5,10,13,16)(9,12,14,15)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,21,23,30)(18,25,20,28)(19,27,26,32)(22,29,24,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(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=g,b^2=h,c^2=ghi,d^2=hi,e^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, 8662610);