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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,8)(5,15)(6,16)(7,9)(11,12)(13,14)(20,28)(24,31)(26,32)(29,30)(33,34,39,43)(35,44,47,40)(36,37,41,48)(38,46,45,42);(3,4)(8,10)(11,13)(12,14)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,47)(34,40,43,44)(36,45,41,38)(37,46,48,42);(1,3,7,4)(2,8,9,10)(5,11,16,13)(6,12,15,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,41)(34,37,43,48)(35,45,47,38)(40,46,44,42);(1,4,7,3)(2,8,9,10)(5,11,16,13)(6,14,15,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,48)(44,47)(45,46);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,41)(35,46)(36,43)(38,40)(39,48)(42,47)(44,45);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,42)(35,36)(37,40)(39,45)(41,47)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,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, 8042868);