Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #23798 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
988 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 988 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,9)(7,11)(10,12)(13,14,17,21,18,16,20,25)(15,19,24,27,22,23,26,28);(5,6,7,9,8,10,11,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,26)(21,27)(25,28);(1,2,3,4)(15,22)(19,23)(24,26)(27,28);(6,10)(9,12)(13,16,17,25,18,14,20,21)(15,23,24,28,22,19,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72318330, #72318392, C2×#212842, … |
| maximal normal subgroup of | #72318330, #72318392, C2×#212842, … |
| maximal quotient of | #72318625, #72318790, C2×#212842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120038);