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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,3)(7,9)(8,10)(15,16)(18,19)(23,25)(24,26)(30,31)(36,44)(39,46)(42,48)(45,47);(2,3)(7,9)(8,10)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=gi,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, 8657923);