Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 956 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 28 points:
(1,2)(3,4)(13,14,17,22,18,23,16,20)(15,21,25,27,26,19,24,28);(5,6,8,10)(7,11,12,9)(14,19,23,21)(20,28,22,27);(2,4)(5,7,8,12)(6,9,10,11)(14,20,23,22)(19,27,21,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ehi,g^b=ghi,d^c=dhi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268602 |
| maximal subgroup of | #72210084, #72261886, #72261949, … |
| maximal normal subgroup of | #72261886, #72261949, #72261960, … |
| maximal quotient of | #72210066, C2×#149485, #72268606 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56681);