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 | #54970 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,6,4,9)(5,13,15,10)(11,14,12,16)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34)(35,43)(36,45)(37,40)(38,44)(39,41)(42,47)(46,48);(1,3)(2,6)(4,7)(5,11)(8,9)(10,16)(12,15)(13,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,39,37)(34,40,41,43)(36,46,47,44)(38,42,48,45);(2,8)(3,4)(10,13)(11,12)(20,29)(23,28)(26,32)(30,31)(34,41)(36,47)(40,43)(44,46);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,14,13,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32)(33,36)(34,42)(35,44)(37,46)(38,40)(39,47)(41,45)(43,48);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,16,9,14)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,37,39,35)(34,43,41,40)(36,46,47,44)(38,42,48,45);(1,6,7,9)(2,4,8,3)(5,14,15,16)(10,11,13,12)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,38,39,48)(34,44,41,46)(35,42,37,45)(36,43,47,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625541);