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 | |
4200 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,28,22,24)(20,26,30,32)(21,29,31,27)(33,34)(35,44)(36,41)(37,40)(38,42)(39,43)(45,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,25,28)(20,27,30,29)(21,26,31,32)(33,35,39,37)(34,40,43,44)(36,45,42,47)(38,46,41,48);(3,10)(6,7)(11,16)(14,15)(18,25)(20,30)(24,28)(27,29)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,48,44,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,30)(18,26,25,32)(19,29,22,27)(21,28,31,24)(33,36)(34,41)(35,45)(37,47)(38,43)(39,42)(40,48)(44,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,27)(19,26)(20,28)(22,32)(23,31)(24,30)(25,29)(33,37,39,35)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,19)(18,24,25,28)(20,29,30,27)(21,26,31,32)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,47,44,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635729);