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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4305 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)(19,28)(20,22)(21,26)(23,25)(24,30)(27,31)(29,32)(33,34,38,40)(35,36,42,44)(37,39,48,47)(41,43,45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22)(33,35)(34,36)(37,45)(38,42)(39,46)(40,44)(41,48)(43,47);(3,10)(6,7)(11,16)(14,15)(18,25)(20,27)(24,28)(26,29)(33,36,38,44)(34,35,40,42)(37,43,48,46)(39,41,47,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24)(33,36)(34,35)(37,46)(38,44)(39,45)(40,42)(41,47)(43,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28)(33,37)(34,39)(35,41)(36,43)(38,48)(40,47)(42,45)(44,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)(33,34)(35,36)(37,47)(38,40)(39,48)(41,46)(42,44)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^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, 9073980);