Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1125 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)(7,16)(8,21)(9,12)(10,15)(11,19)(13,25)(14,26)(17,29)(18,31)(20,28)(22,27)(23,32)(24,30)(33,36)(34,35);(5,7)(6,11)(8,22)(9,17)(10,20)(12,16)(13,31)(14,28)(15,29)(18,23)(19,26)(21,35)(24,34)(25,33)(27,32)(30,36);(6,12)(7,17)(9,26)(13,30)(14,15)(16,28)(18,33)(19,20)(21,32)(22,34)(23,24)(31,35);(5,8,10,25)(6,13,15,21)(7,18)(9,23,26,24)(11,27)(12,30,14,32)(16,31)(17,33)(19,22)(20,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292059);