Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2382 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2382 conjugacy classes of subgroups — too many to draw. Only its 310 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:
(2,7,10,20)(3,12)(4,16)(8,22,25,29)(9,17,11,21)(15,27)(18,28)(19,24,32,23);(1,2)(3,8)(4,17)(5,11)(6,9)(7,18)(10,13)(12,23)(14,29)(15,24)(16,20)(19,31)(21,28)(22,30)(25,27)(26,32);(3,12)(4,18)(7,17)(8,23)(14,30)(15,27)(16,28)(19,32)(20,21)(22,29)(24,25)(26,31);(1,3,13,27)(2,8,10,25)(4,14,28,30)(5,12,6,15)(7,19,20,32)(9,24,11,23)(16,26,18,31)(17,29,21,22)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72235011, #72235013, #72235018, … |
| maximal normal subgroup of | #72235011, #72235013, #72235018, … |
| maximal quotient of | #72239243 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94029);