Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93192 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C2×C4⋊C4 | C2×C8 |
| Frattini, cosocle | #1445 | C22 |
| prime radical | #3492 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1445 ◁ G | |
| LECS | G ▷ #1445 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
268 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,10,11,15,12,16,13,6)(4,7)(5,9)(8,14)(17,18,19,23)(20,29,30,26)(21,25,28,32)(22,31,24,27);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,18)(19,26)(20,29)(21,25)(22,28)(23,27)(24,31)(30,32)Polycyclic presentation:
2^9|a^2=d,c^2=h,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=dehi,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72267807, #72267851 |
| maximal normal subgroup of | #72267807, #72267851 |
| maximal quotient of | #72267812 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388);