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 | |
4152 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)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(33,34)(35,40)(36,41)(37,42)(38,43)(39,46)(44,48)(45,47);(1,3,7,14)(2,8,6,11)(4,5,13,12)(9,10,15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,38,46)(34,39,43,40)(36,47,42,44)(37,45,41,48);(2,6)(4,13)(5,12)(8,11)(18,22)(19,26)(20,29)(31,32)(33,35,38,46)(34,40,43,39)(36,47,42,44)(37,48,41,45);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,38,42)(34,41,43,37)(35,44,46,47)(39,45,40,48);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965390);