Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,22,18,23,15,24)(16,21,25,28,26,19,27,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,19)(16,26)(20,24)(21,23)(22,28)(25,27);(2,4)(5,8,9,7)(6,10,11,12)(13,15,18,17)(14,20,23,28)(16,27,26,25)(19,24,21,22)Polycyclic presentation:
2^9|a^2=g,b^2=dei,c^2=de,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,e^c=ei,g^c=ghi,g^d=gi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56510);