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 | #2431 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3923 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)(3,5,10,8)(4,12,15,9)(11,16,13,14)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,47)(44,46);(2,7)(3,10)(9,12)(11,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,35,39,36)(34,40,42,45)(37,46,41,43)(38,44,48,47);(2,7)(5,8)(9,12)(14,16)(18,23)(21,31)(24,29)(25,32)(33,36,39,35)(34,40,42,45)(37,46,41,43)(38,47,48,44);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,37,39,41)(34,38,42,48)(35,43,36,46)(40,47,45,44);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,38)(34,41)(35,44)(36,47)(37,42)(39,48)(40,43)(45,46);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229316);