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 | #46289 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(33,34)(35,43)(36,39)(37,46)(38,44)(40,45)(41,47)(42,48);(2,3)(7,9)(8,10)(14,15)(19,26)(20,25)(21,30)(23,29)(34,36)(37,47)(40,43)(42,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,35)(34,40)(36,43)(37,44)(38,41)(39,45)(42,47)(46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36,39,34)(35,43,45,40)(37,46,47,41)(38,44,48,42);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,37)(34,41)(35,44)(36,46)(38,40)(39,47)(42,45)(43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,38,39,48)(34,42,36,44)(35,41,45,46)(37,40,47,43)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=g,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,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015930);