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 | |
4136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,9,11,5)(6,12,14,8)(10,13,16,15)(17,18)(19,24)(20,30)(21,25)(22,31)(23,26)(27,28)(29,32)(33,34,40,38)(35,43,39,41)(36,37,42,47)(44,46,48,45);(1,3,7,11)(2,5,4,9)(6,10,14,16)(8,13,12,15)(17,19,23,29)(18,24,26,32)(20,25,31,27)(21,22,28,30)(33,35,40,39)(34,41,38,43)(36,44,42,48)(37,45,47,46);(2,4)(5,9)(6,14)(10,16)(17,20,23,31)(18,22,26,30)(19,27,29,25)(21,32,28,24)(33,36,40,42)(34,37,38,47)(35,44,39,48)(41,45,43,46);(1,4,7,2)(3,9,11,5)(6,12,14,8)(10,15,16,13)(17,20)(18,22)(19,25)(21,32)(23,31)(24,28)(26,30)(27,29)(33,37,40,47)(34,42,38,36)(35,45,39,46)(41,48,43,44);(1,5,7,9)(2,3,4,11)(6,13,14,15)(8,10,12,16)(17,21,23,28)(18,25,26,27)(19,22,29,30)(20,24,31,32)(33,38,40,34)(35,41,39,43)(36,37,42,47)(44,46,48,45);(1,6,7,14)(2,8,4,12)(3,10,11,16)(5,13,9,15)(17,22)(18,20)(19,28)(21,29)(23,30)(24,25)(26,31)(27,32)(33,39,40,35)(34,41,38,43)(36,44,42,48)(37,46,47,45)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961156);