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 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,22,26)(19,23,24,20)(21,25,30,29)(27,31,32,28)(34,40)(35,44)(36,46)(43,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,35)(34,41)(36,45)(37,46)(38,40)(39,44)(42,43)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(33,36,39,46)(34,42,40,47)(35,45,44,37)(38,48,41,43);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,37,39,45)(34,43,40,48)(35,46,44,36)(38,47,41,42);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38)(34,44)(35,40)(36,43)(37,47)(39,41)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8337626);