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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,14,15,9)(10,16,12,13)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(1,3,7,4)(2,5,8,11)(6,10,15,12)(9,13,14,16)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(2,8)(3,4)(6,15)(13,16)(19,25)(20,28)(22,26)(23,30)(34,39)(36,40)(37,46)(42,48);(1,4,7,3)(2,5,8,11)(6,12,15,10)(9,13,14,16)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,48)(46,47);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,14)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,14)(11,16)(17,20,21,28)(18,23,24,30)(19,27,25,31)(22,29,26,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834562);