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 | |
3913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(34,40)(35,37)(42,46)(43,45);(2,6)(7,13)(8,11)(14,16)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=i,c^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, 9388655);