Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 789 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(7,10)(8,18)(9,20)(13,28)(14,29)(15,27)(16,31)(17,24)(19,23)(21,32)(22,33)(25,26)(30,35)(34,36);(5,6)(7,13)(8,19)(9,23)(10,25)(11,15)(12,18)(14,20)(16,29)(17,30)(21,35)(22,34)(24,36)(26,32)(27,33)(28,31);(2,4);(5,7,11,10)(6,13,15,25)(8,16,22,21)(9,17,24,20)(12,26,27,28)(14,23,30,36)(18,32,33,31)(19,29,34,35)Polycyclic presentation:
2^9|d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72286674, #72319630, D4×#2263, … |
| maximal normal subgroup of | #72286674, #72319630, D4×#2263, … |
| maximal quotient of | #72286704, #72321065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420026);