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 | |
4066 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(21,29)(26,30)(27,32)(28,31)(33,34,39,37)(35,41,46,40)(36,43,38,42)(44,47,45,48);(3,9)(7,13)(10,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,40)(36,44)(37,41)(38,45)(39,46)(42,47)(43,48);(2,6)(7,13)(8,11)(14,16)(18,23)(20,24)(26,30)(28,31)(33,35)(34,41)(36,44)(37,40)(38,45)(39,46)(42,48)(43,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,36,39,38)(34,42,37,43)(35,44,46,45)(40,47,41,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37,39,34)(35,41,46,40)(36,43,38,42)(44,48,45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,38,39,36)(34,43,37,42)(35,45,46,44)(40,48,41,47)Polycyclic presentation:
2^9|a^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002671);