Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| 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 | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1459 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)(5,6,11,16)(7,17,21,26)(8,22,25,29)(9,15,10,14)(12,18,28,20)(13,23,30,24)(19,27,33,36)(31,34,35,32);(5,7,9,20,11,21,10,18)(6,12,14,17,16,28,15,26)(8,19,24,34,25,33,23,32)(13,27,22,31,30,36,29,35);(2,4)(7,18)(8,23)(9,10)(12,26)(13,29)(14,15)(17,28)(19,33)(20,21)(22,30)(24,25)(27,36);(5,8)(6,13)(7,19)(9,24)(10,23)(11,25)(12,27)(14,22)(15,29)(16,30)(17,31)(18,32)(20,34)(21,33)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,g^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,g^c=gi,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, 371266);