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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,9,15,16)(10,13,12,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,39,41)(35,42,38,40)(36,37,46,47)(43,44,45,48);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,35,39,38)(34,40,41,42)(36,45,46,43)(37,44,47,48);(2,8)(3,4)(9,16)(10,12)(20,28)(23,31)(27,32)(29,30)(34,41)(36,46)(40,42)(43,45);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,36)(34,37)(35,43)(38,45)(39,46)(40,48)(41,47)(42,44);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37)(34,36)(35,44)(38,48)(39,47)(40,45)(41,46)(42,43);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,16)(11,14)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38,39,35)(34,40,41,42)(36,43,46,45)(37,44,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,f^2=g,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, 8641577);