Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C42 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2289 conjugacy classes of subgroups; only the 472 normal subgroups are shown.
This group has 2289 conjugacy classes of subgroups — too many to draw. Only its 472 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:
(1,2,5,12,7,11,6,22)(3,13,17,28,16,8,18,24)(4,10,19,26,20,9,21,25)(14,23,29,31,15,27,30,32);(1,3,6,18,7,16,5,17)(2,8,12,24,11,13,22,28)(4,14,19,29,20,15,21,30)(9,23,26,32,10,27,25,31);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,23)(10,11)(12,26)(13,27)(15,16)(17,30)(18,29)(22,25)(24,32)(28,31);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,23)(9,11)(12,26)(13,27)(14,16)(17,30)(18,29)(22,25)(24,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,g^c=gi,h^d=hi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72213423, #72213513, #72213555, … |
| maximal normal subgroup of | #72213423, #72213513, #72213555, … |
| maximal quotient of | C2×#356056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263252);