Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,19,20,18,23,17,22)(15,24,27,28,25,21,26,16);(5,7,9,8)(6,10,11,12)(14,20)(16,21)(17,19)(22,23)(24,28)(26,27);(5,8,9,7)(6,10,11,12)(13,15,18,25)(14,21,23,24)(16,22,28,20)(17,26,19,27);(13,16,17,21,18,28,19,24)(14,15,22,26,23,25,20,27);(15,25)(16,28)(21,24)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72307298, #72307381, #72307428, … |
| maximal normal subgroup of | #72307298, #72307381, #72307428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287875);