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 | |
4013 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)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,46,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,41)(36,46)(37,39)(38,45)(40,47)(42,48)(43,44);(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,40,37)(34,42,38,43)(35,46,47,39)(41,48,45,44);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,40,36)(34,43,38,42)(35,46,47,39)(41,48,45,44);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,48,46,44);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,39,40,46)(34,44,38,48)(35,37,47,36)(41,43,45,42)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352894);