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 | #52683 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,11)(3,9,6,7)(4,14,5,10)(12,15,13,16)(17,18,22,26)(19,23,24,20)(21,25,30,29)(27,31,32,28)(33,34,40,38)(35,44,47,41)(36,48,42,37)(39,43,46,45);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35)(34,41)(36,45)(37,39)(38,44)(40,47)(42,43)(46,48);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,36)(34,37)(35,45)(38,48)(39,41)(40,42)(43,47)(44,46);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(33,37,40,48)(34,42,38,36)(35,46,47,39)(41,45,44,43);(1,6,8,3)(2,7,11,9)(4,13,5,12)(10,15,14,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,38,40,34)(35,44,47,41)(36,48,42,37)(39,45,46,43);(1,7)(2,3)(4,15)(5,16)(6,11)(8,9)(10,12)(13,14)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,39,40,46)(34,43,38,45)(35,37,47,48)(36,41,42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,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, 8353146);