Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(6,9)(7,10)(11,12)(14,18)(15,19)(16,22)(20,28)(21,26)(23,27)(24,25);(2,4)(13,14,19,25,16,20,23,26,17,21,27,28,22,24,15,18);(5,6,10,12,8,11,7,9)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72216165, #72216196, #72216411, … |
| maximal normal subgroup of | #72216165, #72216196, #72216411, … |
| maximal quotient of | #72216196, #72216251, #72216288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48861);