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 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(2,6)(7,9)(8,13)(14,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^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, 8062234);