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 | #56535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,26)(19,25,21,24)(20,29,31,22)(27,28,30,32)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,21)(18,24,26,25)(20,30,31,27)(22,32,29,28)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(2,6)(7,10)(8,12)(14,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,32)(26,29)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,25,26,24)(20,30,31,27)(22,32,29,28);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,28)(29,31)(30,32)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663298);