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 | |
3945 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)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(34,35)(40,43)(41,42)(47,48);(3,10)(6,7)(11,16)(14,15)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=g,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, 8635490);