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 | |
4143 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(36,44)(40,45)(42,48)(46,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(18,24)(19,20)(26,29)(27,28)(34,38)(39,41)(40,45)(46,47);(2,7)(3,8)(9,13)(11,15)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=ghi,e^2=h,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=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014400);