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 | #52308 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,48)(46,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35,39,45)(34,40,43,38)(36,44,37,47)(41,46,42,48);(3,12)(6,15)(8,16)(10,13)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,36,39,37)(34,41,43,42)(35,44,45,47)(38,48,40,46);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35)(34,38)(36,44)(37,47)(39,45)(40,43)(41,48)(42,46);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,36)(34,42,43,41)(35,44,45,47)(38,48,40,46);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38)(34,35)(36,46)(37,48)(39,40)(41,47)(42,44)(43,45)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,e^2=g,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, 8314105);