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 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,15)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,40)(35,44)(36,46)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35,39,44)(34,41,40,38)(36,45,46,37)(42,43,47,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36)(34,42)(35,45)(37,44)(38,48)(39,46)(40,47)(41,43);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30)(33,37,39,45)(34,43,40,48)(35,36,44,46)(38,42,41,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,38,39,41)(34,44,40,35)(36,43,46,48)(37,42,45,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352885);