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 | #54510 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,22)(20,21)(23,25)(24,26)(27,30)(28,31)(29,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,23,26)(18,24,25,22)(20,27,29,31)(21,28,32,30)(34,40)(35,43)(41,42)(44,45);(2,7)(3,10)(8,13)(11,16)(18,25)(20,29)(22,24)(27,31)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,29)(18,21,25,32)(19,27,26,31)(22,30,24,28)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,20)(19,28)(22,31)(23,32)(24,27)(25,29)(26,30)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22,23,24)(18,26,25,19)(20,30,29,28)(21,31,32,27)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=gi,e^2=gh,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, 9388727);