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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18)(19,29)(20,22)(21,26)(23,27)(24,28)(25,32)(30,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,3)(4,11)(8,10)(12,16)(17,19,24,32)(18,25,28,29)(20,30,21,27)(22,23,26,31)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(5,13)(8,10)(12,16)(14,15)(17,20,24,21)(18,26,28,22)(19,30,32,27)(23,25,31,29)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,21,24,20)(18,26,28,22)(19,27,32,30)(23,25,31,29)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,22,24,26)(18,21,28,20)(19,31,32,23)(25,30,29,27)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,23,24,31)(18,27,28,30)(19,26,32,22)(20,29,21,25)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324246);