Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,28,20,24)(21,33,35,25)(23,27,37,43)(29,39,31,46)(30,44,32,40)(34,41,47,48)(36,45,38,42);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,24,26,28)(21,34,35,47)(23,32,37,30)(25,41,33,48)(27,44,43,40)(29,38,31,36)(39,45,46,42);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,19)(18,24,26,28)(21,31,35,29)(23,32,37,30)(25,39,33,46)(27,40,43,44)(34,36,47,38)(41,45,48,42);(10,14)(13,16)(17,21,22,35)(18,25,26,33)(19,29,20,31)(23,38,37,36)(24,39,28,46)(27,45,43,42)(30,47,32,34)(40,48,44,41);(1,4,5,3)(2,7,6,8)(18,26)(19,20)(25,33)(27,43)(29,31)(30,32)(34,47)(42,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7435890);