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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,6,9,15)(4,5,8,14)(11,12,13,16)(17,18)(19,28)(20,22)(21,31)(23,27)(24,30)(25,29)(26,32)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,19,23,30)(18,24,27,28)(20,21,29,32)(22,26,25,31)(34,39)(35,42)(41,44)(43,48);(3,9)(6,15)(11,13)(12,16)(17,20,23,29)(18,25,27,22)(19,21,30,32)(24,31,28,26)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,19)(18,24)(20,21)(22,26)(23,30)(25,31)(27,28)(29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,21)(18,26)(19,29)(20,30)(22,28)(23,32)(24,25)(27,31)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,22,23,25)(18,20,27,29)(19,26,30,31)(21,28,32,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231311);