Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63040 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2568 conjugacy classes of subgroups; only the 304 normal subgroups are shown.
This group has 2568 conjugacy classes of subgroups — too many to draw. Only its 304 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:
(1,2)(3,14)(4,9,15,16)(5,10,6,11)(7,12)(8,13);(1,3,8,7)(2,9,13,16)(4,5,15,6)(10,14,11,12);(1,4,8,15)(2,10,13,11)(3,6,7,5)(9,12,16,14);(1,5,8,6)(2,10,13,11)(3,15,7,4)(9,14,16,12)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234102, #72235439, #72258948, … |
| maximal normal subgroup of | #72234102, #72258948, #72261708, … |
| maximal quotient of | #72234065, #72235343, #72260922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400856);