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 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,9)(8,13)(10,12)(20,28)(23,30)(26,32)(29,31)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,2,6,5)(3,7,9,11)(4,8,14,13)(10,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(3,9)(7,11)(10,12)(15,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,3)(2,7)(4,12)(5,11)(6,9)(8,16)(10,14)(13,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,4,6,14)(2,8,5,13)(3,10,9,12)(7,15,11,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,5,6,2)(3,11,9,7)(4,13,14,8)(10,16,12,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064499);