Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16)(7,10,15,11)(8,19,24,17)(9,18)(13,23,20,22)(14,27)(21,33)(25,30)(26,32)(28,31)(29,36)(34,35);(1,2)(3,4)(5,7,10,16,12,15,11,6)(8,20,22,17,24,13,23,19)(9,18,27,32,28,31,26,14)(21,33,35,30,36,29,25,34);(2,4)(5,8,12,24)(6,13,16,20)(7,17,15,19)(9,25,28,35)(10,23,11,22)(14,30,32,34)(18,33,31,29)(21,27,36,26);(5,9)(6,14)(7,18)(8,21)(10,27)(11,26)(12,28)(13,29)(15,31)(16,32)(17,30)(19,34)(20,33)(22,35)(23,25)(24,36)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 363707);