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 | #47057 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4058 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,29)(21,26)(22,25)(23,24)(28,31)(30,32)(33,34)(35,46)(36,42)(37,41)(38,43)(39,44)(40,45)(47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,28)(22,32)(23,27)(24,26)(25,31)(29,30)(33,35,40,37)(34,41,45,46)(36,38,47,44)(39,48,43,42);(3,10)(6,7)(11,16)(14,15)(17,20,24,22)(18,25,23,29)(19,28,26,32)(21,31,27,30)(33,36,40,47)(34,42,45,48)(35,38,37,44)(39,46,43,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,24,27)(18,26,23,19)(20,30,22,31)(25,28,29,32)(33,37,40,35)(34,41,45,46)(36,38,47,44)(39,42,43,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,24,20)(18,25,23,29)(19,28,26,32)(21,30,27,31)(33,38)(34,43)(35,47)(36,37)(39,45)(40,44)(41,42)(46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,23,24,18)(19,27,26,21)(20,29,22,25)(28,30,32,31)(33,39,40,43)(34,44,45,38)(35,48,37,42)(36,46,47,41)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074809);