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 | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3898 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)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(18,20)(23,26)(24,28)(30,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(19,25)(23,26)(27,31)(30,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(4,12)(7,15)(11,16)(13,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,4)(2,7)(3,11)(5,14)(6,12)(8,13)(9,16)(10,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646474);