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 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,14)(9,11)(10,16)(12,15)(20,28)(24,29)(26,32)(30,31)(33,34,39,43)(35,42,38,40)(36,37,47,41)(44,45,46,48);(3,9)(4,11)(10,15)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,40,43,42)(36,44,47,46)(37,48,41,45);(2,7)(3,9)(8,13)(10,15)(18,22)(23,25)(24,29)(30,31)(33,36)(34,41)(35,44)(37,43)(38,46)(39,47)(40,45)(42,48);(1,3)(2,4)(5,10)(6,9)(7,11)(8,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,41)(34,36,43,47)(35,45,38,48)(40,46,42,44);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,39,38)(34,42,43,40)(36,44,47,46)(37,45,41,48);(1,5,6,14)(2,8,7,13)(3,10,9,15)(4,12,11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,40,43,42)(36,46,47,44)(37,48,41,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000670);