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 | |
3949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(18,24)(19,20)(26,29)(27,28)(34,40)(35,43)(41,42)(44,45);(3,11)(7,8)(12,16)(13,14)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=g,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, 8638401);