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 | #2661 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3931 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(34,35)(39,41)(40,42)(46,47);(2,7)(3,8)(9,13)(11,16)(19,28)(20,29)(24,31)(25,30)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,2)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625632);