Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92835 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1744 | C22 |
| prime radical | #3169 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ #1744 ▷ C2×C42 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 198 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:
(1,2,4,10,7,12,15,16,5,13,14,8,6,11,3,9)(17,18,21,26,23,27,32,25,24,28,20,31,22,19,29,30);(2,8)(3,14)(4,15)(6,7)(9,11)(10,13)(12,16)(17,19)(18,23)(20,25)(21,30)(22,28)(24,27)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=g,e^2=i,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210224, #72211373, C2×#92837, … |
| maximal normal subgroup of | #72211373, C2×#92837, #72213171, … |
| maximal quotient of | #72210220, #72210670, #72210671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33);