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 | #2703 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3809 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)(3,9)(4,8)(5,13)(6,10)(7,15)(11,14)(12,16)(18,20)(23,26)(24,28)(30,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3)(2,5)(4,11)(6,13)(7,14)(8,16)(9,10)(12,15)(19,25)(23,26)(27,31)(30,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(1,4,6,15)(2,7,10,8)(3,11,13,12)(5,14,9,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,4)(2,8)(3,12)(5,14)(6,15)(7,10)(9,16)(11,13)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,5)(2,3)(4,14)(6,9)(7,11)(8,12)(10,13)(15,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,3)(2,9)(4,11)(5,10)(6,13)(7,16)(8,14)(12,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gh,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, 8646498);