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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,7)(3,11,6,9)(4,10,5,14)(12,16,13,15)(18,23)(19,28)(26,30)(27,32)(34,39)(35,42)(36,44)(45,47);(1,3,8,6)(2,9,7,11)(4,12,5,13)(10,15,14,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(1,4,8,5)(2,10,7,14)(3,12,6,13)(9,15,11,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,5,8,4)(2,10,7,14)(3,13,6,12)(9,15,11,16)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6,8,3)(2,11,7,9)(4,13,5,12)(10,16,14,15)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,7,8,2)(3,11,6,9)(4,14,5,10)(12,16,13,15)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343852);