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 |
| prime radical | #57183 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(9,14)(10,15)(11,16)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(2,7)(3,8)(9,14)(10,15)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019752);