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 | #53610 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,11,13,8)(5,14,12,9)(6,10,16,15)(17,18)(19,21)(20,25)(22,26)(23,27)(24,30)(28,31)(29,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,4)(8,11)(9,14)(10,15)(17,20,23,29)(18,25,27,32)(19,28,30,22)(21,31,24,26)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,2)(3,8,13,11)(5,14,12,9)(6,10,16,15)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,26)(27,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,21,23,24)(18,19,27,30)(20,31,29,26)(22,25,28,32)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,22,23,28)(18,26,27,31)(19,29,30,20)(21,32,24,25)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646638);