Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #247017 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
1325 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1325 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)(3,13)(4,11)(5,10)(6,24)(7,15)(8,14)(9,17)(12,21)(16,25)(18,29)(19,28)(20,30)(22,32)(23,26)(27,31);(1,3)(2,9)(4,19)(5,6)(7,18)(8,16)(10,26)(11,13)(12,29)(14,27)(15,28)(17,21)(20,22)(23,31)(24,32)(25,30);(2,10)(3,5)(4,20)(9,12)(11,28)(13,24)(14,29)(15,25)(16,18)(17,19)(21,23)(22,31)(26,27)(30,32);(2,11)(3,16)(4,21)(5,18)(9,25)(10,28)(12,15)(13,27)(14,30)(17,22)(19,31)(20,23)(24,26)(29,32)Polycyclic presentation:
2^9|e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213691, #72219528, #72286478, … |
| maximal normal subgroup of | #72213691, #72219528, #72286478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154213);