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 | #50733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,5,11,7)(4,9,13,16)(10,14,12,15)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34,40,42)(35,38,39,36)(37,41,47,48)(43,46,45,44);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,40,39)(34,36,42,38)(37,45,47,43)(41,46,48,44);(2,8)(4,13)(5,7)(10,12)(19,27)(20,30)(23,31)(24,29)(33,36,40,38)(34,35,42,39)(37,46,47,44)(41,45,48,43);(1,4)(2,9)(3,10)(5,14)(6,13)(7,15)(8,16)(11,12)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,40,47)(34,41,42,48)(35,43,39,45)(36,44,38,46);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,13)(15,16)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,38,40,36)(34,35,42,39)(37,44,47,46)(41,45,48,43);(1,5,6,7)(2,3,8,11)(4,14,13,15)(9,10,16,12)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,39,40,35)(34,36,42,38)(37,45,47,43)(41,44,48,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,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, 8841513);