Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,8,11,9,12,7,10)(13,14)(15,21)(16,24,25,19)(17,20)(18,23,26,28)(22,27);(1,2,3,4)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,27)(25,28);(5,7,9,8)(6,10,12,11)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,e^2=h,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282597, #72282631, #72282655 |
| maximal normal subgroup of | #72282597, #72282631, #72282655 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34144);