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 | |
3944 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)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,39,43)(35,44,46,40)(36,41,45,37)(38,42,47,48);(2,8)(3,4)(10,13)(11,12)(19,28)(20,29)(24,30)(25,32)(33,35,39,46)(34,40,43,44)(36,38,45,47)(37,42,41,48);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,24)(20,21)(22,25)(23,28)(26,29)(27,30)(31,32)(33,36,39,45)(34,41,43,37)(35,38,46,47)(40,48,44,42);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,20)(18,25)(19,21)(22,24)(23,29)(26,28)(27,32)(30,31)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,21)(18,22)(19,20)(23,26)(24,25)(27,31)(28,29)(30,32)(33,37)(34,36)(35,42)(38,40)(39,41)(43,45)(44,47)(46,48);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,22)(18,26)(19,25)(20,30)(21,27)(23,31)(24,29)(28,32)(33,38)(34,42)(35,45)(36,46)(37,44)(39,47)(40,41)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8599759);