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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3980 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)(3,8,13,16)(4,5,9,14)(6,15,12,10)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,47)(45,46);(1,3,7,13)(2,8,11,16)(4,12,9,6)(5,10,14,15)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,4,7,9)(2,5,11,14)(3,6,13,12)(8,15,16,10)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,45,42,47);(1,4)(2,5)(3,12)(6,13)(7,9)(8,10)(11,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,37,39,48)(34,41,43,36)(35,45,38,47)(40,46,42,44);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,16)(13,15)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,48)(44,45)(46,47);(1,6,7,12)(2,10,11,15)(3,4,13,9)(5,8,14,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081503);