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 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,23,27)(19,28,30,24)(20,26,31,25)(21,32,29,22)(33,34,39,37)(35,43,46,40)(36,41,45,47)(38,42,44,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,24)(20,21)(22,26)(23,30)(25,32)(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)(17,20,23,31)(18,25,27,26)(19,21,30,29)(22,24,32,28)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,26)(19,29)(21,30)(22,28)(23,31)(24,32)(25,27)(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)(17,21,23,29)(18,22,27,32)(19,20,30,31)(24,26,28,25)(33,37,39,34)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22)(18,21)(19,26)(20,24)(23,32)(25,30)(27,29)(28,31)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)Polycyclic presentation:
2^9|a^2=g,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, 8635883);