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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,7,10,8)(4,9,12,13)(11,14,16,15)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34,39,38)(35,44,46,40)(36,42,47,41)(37,48,45,43);(1,3)(2,7)(4,11)(5,8)(6,10)(9,14)(12,16)(13,15)(19,26)(22,25)(27,28)(31,32)(33,35)(34,40)(36,37)(38,44)(39,46)(41,48)(42,43)(45,47);(3,10)(7,8)(11,16)(14,15)(17,19,21,26)(18,22,24,25)(20,27,30,28)(23,31,29,32)(33,36,39,47)(34,41,38,42)(35,37,46,45)(40,48,44,43);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,36)(34,42)(35,45)(37,46)(38,41)(39,47)(40,48)(43,44);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,37,39,45)(34,43,38,48)(35,36,46,47)(40,42,44,41);(1,5,6,2)(3,8,10,7)(4,9,12,13)(11,14,16,15)(17,20,21,30)(18,23,24,29)(19,27,26,28)(22,31,25,32)(33,38,39,34)(35,44,46,40)(36,42,47,41)(37,43,45,48)Polycyclic presentation:
2^9|a^2=gi,c^2=h,e^2=g,f^2=gi,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, 8653953);