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 | |
4005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,8,4)(5,9,14,15)(11,16,13,12)(18,22)(20,28)(23,25)(26,32)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3)(2,4)(5,13)(6,10)(7,8)(9,16)(11,14)(12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4,7,10)(2,8,6,3)(5,12,14,16)(9,13,15,11)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,14)(8,13)(10,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48);(1,6,7,2)(3,10,8,4)(5,9,14,15)(11,12,13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9701484);