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 | #53472 | 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,23)(19,26)(21,29)(28,31)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35,39,46)(34,40,36,43)(37,38,45,44)(41,48,47,42);(2,6)(7,13)(8,11)(14,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,34,39,36)(35,40,46,43)(37,41,45,47)(38,48,44,42);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,39,34)(35,43,46,40)(37,41,45,47)(38,48,44,42);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388681);