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 | |
4096 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)(20,25)(21,32)(22,27)(23,30)(24,31)(26,29)(33,34)(35,37)(36,41)(38,42)(39,43)(40,46)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,21)(22,30)(24,32)(25,26)(27,28)(29,31)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(3,10)(6,7)(11,16)(14,15)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,32,28,26)(33,36,39,45)(34,41,43,48)(35,44,46,38)(37,47,40,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,30)(22,31)(23,32)(24,27)(26,28)(33,36)(34,41)(35,44)(37,47)(38,46)(39,45)(40,42)(43,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,25,28,24)(33,37,39,40)(34,35,43,46)(36,47,45,42)(38,41,44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,18)(19,23)(20,24)(21,32)(22,27)(25,31)(26,29)(28,30)(33,38,39,44)(34,42,43,47)(35,45,46,36)(37,48,40,41)Polycyclic presentation:
2^9|c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635882);