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) = 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)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30)(33,34)(35,41)(36,38)(37,40)(39,42)(43,47)(44,45)(46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,40,42,41)(36,43,46,45)(38,44,48,47);(3,10)(6,7)(11,16)(14,15)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,41,42,40)(36,45,46,43)(38,44,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,36,39,46)(34,38,42,48)(35,43,37,45)(40,44,41,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31)(33,37,39,35)(34,40,42,41)(36,45,46,43)(38,44,48,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)(33,38)(34,36)(35,44)(37,47)(39,48)(40,43)(41,45)(42,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=gh,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, 8635459);