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 | #56272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4010 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,40,44)(35,45,38,41)(36,42,37,46)(39,43,48,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35,40,38)(34,41,44,45)(36,43,37,47)(39,46,48,42);(2,6)(7,13)(8,11)(14,16)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,40,37)(34,42,44,46)(35,43,38,47)(39,45,48,41);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,37,40,36)(34,42,44,46)(35,43,38,47)(39,41,48,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,38,40,35)(34,41,44,45)(36,47,37,43)(39,46,48,42);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,39)(34,43)(35,46)(36,45)(37,41)(38,42)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10153982);