Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92825 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | #1744 | C22 |
| prime radical | C2×#127 | C8 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ #1744 ▷ C2×C42 ▷ C22 ▷ C1 | |
226 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16)(17,18,19,25,21,23,27,30,22,26,29,32,20,24,28,31);(2,6)(8,13)(18,23)(19,27)(20,21)(24,26)(25,32)(28,29)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,e^2=i,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,g^b=gi,d^c=dhi56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#92827, #72213171, #72213174, … |
| maximal normal subgroup of | C2×#92827, #72213171, #72213174, … |
| maximal quotient of | #72210668, #72210670, #72210672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23);