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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(34,39)(40,43)(41,42)(47,48);(4,12)(9,14)(11,13)(15,16)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,47)(42,48);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,36,38,37)(34,41,39,42)(35,44,46,45)(40,47,43,48);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,37,38,36)(34,42,39,41)(35,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646589);