Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149765 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11503 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 16 points:
(1,2,3,8,4,9,11,16,5,6,10,13,12,7,14,15);(2,6)(3,10)(4,12)(8,13);(2,7,6,9)(3,10)(8,16,13,15)(11,14)Polycyclic presentation:
2^9|a^2=e,c^2=h,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=ef,g^b=gi,e^c=ei,e^d=eh,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72272670, #72272684, #72272686 |
| maximal normal subgroup of | #72272670, #72272684, #72272686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56961);