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 | |
4131 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(20,28)(24,29)(26,32)(30,31)(33,34,38,42)(35,41,37,39)(36,40,45,48)(43,47,44,46);(3,11)(4,13)(8,14)(9,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(2,7)(3,11)(9,16)(10,12)(18,22)(23,25)(24,29)(30,31)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000648);