Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #331 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,19)(15,23)(16,17)(20,27)(21,22)(24,25);(6,9,11,10)(13,14,15,20,17,22,24,28,18,21,25,27,16,19,23,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93382, #72214396, #72214402, … |
| maximal normal subgroup of | C2×#93382, #72214396, #72214402, … |
| maximal quotient of | #72210667, #72210668, #72210669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 578);