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 | #2676 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(11,13)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,40)(35,45)(36,44)(43,47);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,39,40)(35,41,45,48)(36,38,44,42)(37,43,46,47);(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35,39,45)(34,41,40,48)(36,46,44,37)(38,47,42,43);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36)(34,42)(35,46)(37,45)(38,40)(39,44)(41,43)(47,48);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30)(33,37,39,46)(34,43,40,47)(35,36,45,44)(38,48,42,41);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,38,39,42)(34,44,40,36)(35,47,45,43)(37,48,46,41)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352887);