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 | |
3915 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(3,9)(7,13)(10,15)(14,16)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(2,6)(7,13)(8,11)(14,16)(18,24)(21,30)(23,26)(27,28)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,26)(21,25)(22,24)(27,31)(28,32)(29,30)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388653);