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 | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,6,10,15)(4,14,13,8)(5,9,12,16)(17,18,23,27)(19,28,31,24)(20,22,26,32)(21,30,29,25)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,7,10)(2,6,11,15)(4,5,13,12)(8,16,14,9)(17,19,23,31)(18,24,27,28)(20,29,26,21)(22,30,32,25)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,7,13)(2,8,11,14)(3,12,10,5)(6,9,15,16)(17,20,23,26)(18,22,27,32)(19,29,31,21)(24,30,28,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,12)(5,13)(7,10)(8,9)(11,15)(14,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,32)(28,31)(29,30)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(1,5,7,12)(2,9,11,16)(3,13,10,4)(6,8,15,14)(17,21)(18,25)(19,20)(22,24)(23,29)(26,31)(27,30)(28,32);(1,6)(2,10)(3,11)(4,9)(5,8)(7,15)(12,14)(13,16)(17,22)(18,26)(19,30)(20,27)(21,24)(23,32)(25,31)(28,29)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,e^2=i,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, 8352844);