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 | #56736 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,13,15)(4,10,14,16)(5,6,12,11)(19,20)(23,24)(27,28)(30,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3)(2,8)(4,5)(6,10)(7,13)(9,15)(11,16)(12,14)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(34,39)(35,42)(41,44)(43,48);(2,9)(3,13)(5,12)(10,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,10)(3,12)(5,13)(6,15)(7,14)(8,11)(9,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,7,12)(2,11,9,6)(3,4,13,14)(8,16,15,10)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,6,7,11)(2,12,9,5)(3,10,13,16)(4,8,14,15)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440803);