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 | #42658 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3865 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)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(18,22)(19,26)(24,28)(27,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(3,12)(6,13)(8,16)(10,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325351);