Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 201326592 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18117 conjugacy classes of subgroups; only the 1074 normal subgroups are shown.
This group has 18117 conjugacy classes of subgroups — too many to draw. Only its 1074 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 — 5 generators on 28 points:
(2,4)(6,8)(10,12)(16,24)(19,26)(22,28)(25,27);(5,6)(7,8)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(9,10)(11,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249412, #72249418, #72249442, … |
| maximal normal subgroup of | #72249412, #72249418, #72249442, … |
| maximal quotient of | #72334966, #72334967, #72335075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215224);