Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14,15,19,17,22,23,21)(16,25,24,20,27,18,26,28);(5,7)(6,9)(8,10)(11,12)(14,18)(15,23)(16,26)(19,20)(21,28)(22,25)(24,27);(2,4)(6,7)(10,11)(13,15,17,23)(14,19,22,21)(16,24,27,26)(18,28,25,20)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,f^b=fi,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225044, #72225053, #72225064, … |
| maximal normal subgroup of | #72225044, #72225053, #72225064, … |
| maximal quotient of | #72225034, #72225035, #72225040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36751);