Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2952 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 2952 conjugacy classes of subgroups — too many to draw. Only its 252 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)(13,14)(15,21,23,18)(16,24,25,19)(17,20)(22,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(6,9)(10,12)(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28);(2,4)(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=ef,d^2=g,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72260908, #72318228, #72318448 |
| maximal normal subgroup of | #72260908, #72318228, #72318448 |
| maximal quotient of | #72260934, #72261587, #72320125 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311436);