Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2819 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,10,9,12,7,11)(13,14)(15,23,24,27)(16,21,17,19)(18,22)(20,25)(26,28);(2,4)(6,10)(7,8)(11,12)(14,19)(15,24)(20,23)(21,22)(25,26)(27,28);(13,15)(14,20,22,28)(16,26)(17,25)(18,24)(19,27,21,23)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72282573, #72282732 |
| maximal normal subgroup of | #72282573, #72282732 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38573);