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 | #47918 | C2 |
| 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:
(1,2)(3,8)(4,10)(5,13)(6,7)(9,14)(11,12)(15,16)(19,25)(20,28)(23,26)(24,29)(33,34,39,43)(35,40,44,47)(36,45,41,37)(38,46,48,42);(2,7)(3,10)(9,13)(11,16)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35)(34,40)(36,46)(37,38)(39,44)(41,42)(43,47)(45,48);(2,7)(3,10)(5,14)(12,15)(18,22)(19,25)(24,29)(27,30)(33,36,39,41)(34,37,43,45)(35,42,44,46)(38,40,48,47);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36)(34,37)(35,42)(38,47)(39,41)(40,48)(43,45)(44,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37)(34,41)(35,38)(36,43)(39,45)(40,42)(44,48)(46,47);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38)(34,42)(35,37)(36,47)(39,48)(40,41)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,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, 8840294);