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 | #46405 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4071 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)(4,13)(5,12)(6,7)(9,14)(10,15)(11,16)(19,25)(20,28)(23,26)(24,29)(34,37)(40,42)(41,45)(47,48);(2,7)(3,11)(9,13)(10,12)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,7)(5,15)(8,16)(9,13)(18,22)(23,26)(24,29)(31,32)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^9|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,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, 8839930);