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 | C2×#121 | C23 |
| 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:
(1,2,7,11)(3,8,10,5)(4,13,6,9)(12,16,14,15)(17,18,23,20)(19,22,27,24)(21,25,29,28)(26,30,32,31)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,10)(2,8,11,5)(4,14,6,12)(9,16,13,15)(21,29)(25,28)(26,32)(30,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(1,4,7,6)(2,9,11,13)(3,12,10,14)(5,16,8,15)(17,19,23,27)(18,24,20,22)(21,26,29,32)(25,31,28,30)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,5,7,8)(2,10,11,3)(4,15,6,16)(9,12,13,14)(17,20,23,18)(19,22,27,24)(21,28,29,25)(26,30,32,31)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,6,7,4)(2,9,11,13)(3,12,10,14)(5,15,8,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,2)(3,8)(4,13)(5,10)(6,9)(7,11)(12,16)(14,15)(17,22)(18,19)(20,27)(21,30)(23,24)(25,26)(28,32)(29,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839297);