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 | #2699 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3871 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,25)(19,24,28,23)(20,29,30,21)(26,27,32,31)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,44,47,46);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,35,38,45)(34,39,42,48)(36,43,41,47)(37,44,40,46);(2,3)(7,9)(8,10)(15,16)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,36,38,41)(34,40,42,37)(35,43,45,47)(39,46,48,44);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,26)(21,31)(22,28)(23,25)(27,29)(30,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20,22,30)(18,21,25,29)(19,26,28,32)(23,27,24,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,20)(19,27)(22,29)(23,26)(24,32)(25,30)(28,31)(33,37,38,40)(34,41,42,36)(35,44,45,46)(39,47,48,43)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,e^2=g,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, 8652986);