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 | |
4218 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)(3,8)(4,6)(5,12)(7,10)(9,13)(11,14)(15,16)(20,28)(24,31)(26,32)(29,30)(34,40)(36,44)(38,41)(43,46);(3,10)(7,8)(11,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,34,39,40)(35,41,37,38)(36,42,44,47)(43,48,46,45);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,37)(34,41,40,38)(36,43,44,46)(42,48,47,45);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,42)(35,43)(37,46)(38,45)(39,44)(40,47)(41,48);(1,4,6,2)(3,8,10,7)(5,12,13,9)(11,15,16,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37,39,35)(34,38,40,41)(36,43,44,46)(42,48,47,45);(1,5,6,13)(2,9,4,12)(3,11,10,16)(7,14,8,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,35)(36,45)(37,40)(39,41)(42,43)(44,48)(46,47)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044163);