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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,13,14,8)(10,15,16,12)(17,18)(19,24)(20,22)(21,28)(23,26)(25,29)(27,30)(31,32)(36,44)(40,47)(42,48)(45,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,13)(14,16)(19,26)(23,24)(27,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20,22,18)(19,24,26,23)(21,28,29,25)(27,31,32,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,21,22,29)(18,25,20,28)(19,27,26,32)(23,30,24,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044182);