Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1378 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,10,8,9,11,12)(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);(6,9)(10,12)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=h,b^a=bd,c^a=ch,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236808, #72236810, #72236814, … |
| maximal normal subgroup of | #72236808, #72236810, #72236814, … |
| maximal quotient of | #72237482, #72268610, #72285529 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34169);