Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346294 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2996 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 2996 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 28 points:
(1,2)(3,4)(9,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21);(5,7)(6,9)(8,11)(10,12);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209761, #72209840, #72209872, … |
| maximal normal subgroup of | #72209840, #72209872, #72227592, … |
| maximal quotient of | #72209757, #72209776, #72227595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253504);