Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94419 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7304 |
| commutator | C22×Q8 | C42 |
| Frattini, prime radical, cosocle | C2×#535 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,13,15,28)(4,18,20,27)(5,11,6,10)(7,12)(8,14,23,31)(9,21)(16,29,17,30)(19,24,32,25)(22,26);(1,3,5,15,7,17,6,16)(2,8,10,23,12,25,11,24)(4,19,21,32,20,14,22,31)(9,13,18,29,26,30,27,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=df,h^b=hi,d^c=dhi,f^c=fi,e^d=ei,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72213445, #72216935, #72216942, … |
| maximal normal subgroup of | #72213445, #72216935, #72216942, … |
| maximal quotient of | C2×#94420, #72338274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1616);