Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246957 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1418 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,8)(5,9)(6,12)(7,13)(10,16)(11,17)(15,27)(18,25)(19,26)(20,29)(21,32)(22,30)(23,28)(24,31);(2,8)(3,15)(4,17)(5,20)(9,14)(10,27)(11,12)(13,26)(16,23)(18,21)(19,22)(25,28)(29,32)(30,31);(2,9)(4,19)(5,16)(8,25)(10,13)(11,14)(12,28)(17,22)(20,23)(26,32)(27,30)(29,31);(1,3)(2,10)(4,20)(5,22)(6,18)(7,15)(8,14)(9,13)(11,25)(12,29)(16,17)(19,23)(21,24)(26,27)(28,31)(30,32)Polycyclic presentation:
2^9|e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213685, #72216601, C2×#246957, … |
| maximal normal subgroup of | #72213685, #72216601, C2×#246957, … |
| maximal quotient of | C2×#246957, #72240340, #72240343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154153);