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 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(17,18)(19,23)(20,28)(21,25)(22,26)(24,31)(27,30)(29,32)(33,34,39,43)(35,40,44,47)(36,45,41,37)(38,46,48,42);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,18,22,26)(19,21,25,23)(20,28,31,24)(27,32,29,30)(33,35)(34,40)(36,46)(37,38)(39,44)(41,42)(43,47)(45,48);(2,6)(3,9)(8,13)(11,14)(19,25)(21,23)(27,29)(30,32)(33,36,39,41)(34,37,43,45)(35,42,44,46)(38,40,48,47);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,19,22,25)(18,23,26,21)(20,29,31,27)(24,30,28,32)(33,36)(34,37)(35,42)(38,47)(39,41)(40,48)(43,45)(44,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,20,22,31)(18,24,26,28)(19,27,25,29)(21,30,23,32)(33,37)(34,41)(35,38)(36,43)(39,45)(40,42)(44,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,21,22,23)(18,25,26,19)(20,30,31,32)(24,29,28,27)(33,38)(34,42)(35,37)(36,47)(39,48)(40,41)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840291);