Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6342371 |
| automorphism | Aut(G) order 150994944 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
9882 conjugacy classes of subgroups; only the 1258 normal subgroups are shown.
This group has 9882 conjugacy classes of subgroups — too many to draw. Only its 1258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(10,12)(14,15);(13,14,16,15);(9,10,11,12)(13,15,16,14);(5,6)(7,8);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^9|b^2=f,c^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ei152 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2847, #120×#127, #72234124, … |
| maximal normal subgroup of | D4×#2847, #120×#127, #72234124, … |
| maximal quotient of | #72320282, C2×#6342381, #72334308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249577);