Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,21)(9,16)(10,14)(11,15)(12,19)(13,23)(18,27)(20,30)(22,29)(24,28)(25,26)(31,32)(33,35)(34,36);(5,7)(6,12,15,26)(8,18)(9,19)(10,25)(11,20)(13,27,29,35)(14,17,16,30)(21,31,28,36)(22,32)(23,34)(24,33);(2,4)(8,22)(9,10)(12,26)(13,28)(14,16)(18,32)(19,25)(21,29)(23,24)(27,31)(33,34)(35,36);(5,8,11,24)(6,13,15,29)(7,18,20,33)(9,23,10,22)(12,27,26,35)(14,21,16,28)(17,31,30,36)(19,34,25,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371847);