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 | |
4068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,40,46)(35,47,48,41)(36,43,37,42)(38,39,45,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,41)(36,45)(37,38)(39,43)(40,48)(42,44)(46,47);(5,13)(8,10)(11,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,36,40,37)(34,42,46,43)(35,45,48,38)(39,41,44,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,40,36)(34,43,46,42)(35,45,48,38)(39,41,44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,40,45)(34,44,46,39)(35,37,48,36)(41,42,47,43);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,39,40,44)(34,45,46,38)(35,43,48,42)(36,47,37,41)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8198863);