Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| 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×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,28)(17,22)(18,23)(19,24)(20,26)(21,27);(6,9)(10,12)(13,15,18,16,19,27,17,26)(14,20,22,21,24,28,23,25);(5,7,8,11)(6,10,9,12)(13,16,17,15,19,26,18,27)(14,21,23,20,24,25,22,28);(1,3,4,2)(13,14)(15,20)(16,21)(17,23)(18,22)(19,24)(25,26)(27,28);(13,17,19,18)(14,22,24,23)(15,26,27,16)(20,21,28,25)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224708, #72224717, #72226669 |
| maximal normal subgroup of | #72224708, #72224717, #72226669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280734);