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 | #46526 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,12,13,8)(11,14,16,15)(19,26)(22,24)(27,32)(29,31)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(2,5)(4,13)(7,9)(11,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(3,10)(7,9)(11,16)(14,15)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,3,6,10)(2,7,5,9)(4,11,13,16)(8,14,12,15)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,4,6,13)(2,8,5,12)(3,11,10,16)(7,14,9,15)(17,21,23,30)(18,25,20,28)(19,27,26,32)(22,29,24,31)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,5,6,2)(3,9,10,7)(4,12,13,8)(11,15,16,14)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=ghi,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, 8662614);