Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35001 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1972 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1972 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6,9,12)(7,11,8,10)(14,19)(15,23)(16,17)(21,22)(24,25)(26,27);(2,4)(6,10)(7,8)(11,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,27)(22,28);(5,7,9,8)(6,10,12,11)(13,16,18,17)(14,19,22,21)(15,24,25,23)(20,27,28,26)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249963, #72249986, #72249998, … |
| maximal normal subgroup of | #72249963, #72249986, #72249998, … |
| maximal quotient of | #72250471, #72250528, #72250607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139495);