Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C24 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,8)(4,13)(5,12)(9,16)(10,14)(11,15);(1,2)(3,11)(4,14)(5,15)(6,9)(7,10)(8,13)(12,16);(1,2)(3,9)(4,15)(5,14)(6,11)(7,10)(8,12)(13,16);(1,3,7,6)(2,9,10,11)(4,12,13,5)(8,16,14,15)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gh,d^b=di,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72209905, #72231245, #72231248, … |
| maximal normal subgroup of | #72209905, #72231245, #72231248, … |
| maximal quotient of | #72213426, #72227843, #72228005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420096);