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 | |
4274 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,23,20)(19,22,27,24)(21,25,30,28)(26,29,32,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,27)(18,24,20,22)(21,26,30,32)(25,31,28,29)(33,35)(34,39)(36,42)(37,43)(38,40)(41,48)(44,45)(46,47);(3,11)(4,12)(6,16)(8,13)(18,20)(22,24)(25,28)(29,31)(33,35,38,40)(34,36,42,39)(37,46,47,43)(41,48,45,44);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,23,18)(19,22,27,24)(21,25,30,28)(26,31,32,29)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22)(18,19)(20,27)(21,29)(23,24)(25,26)(28,32)(30,31)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^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, 8834184);