Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188062 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3188 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 3188 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:
(1,2)(3,11,14,7)(4,16,18,8)(5,9)(6,10)(12,26,28,20)(13,22,15,21)(17,31,32,23)(19,24)(25,29,30,27);(1,3)(2,7)(4,12)(5,14)(6,15)(8,20)(9,11)(10,22)(13,19)(16,30)(17,29)(18,27)(21,24)(23,25)(26,31)(28,32);(1,4)(2,8)(3,12)(5,17)(6,18)(7,20)(9,23)(10,16)(11,25)(13,28)(14,29)(15,27)(19,32)(21,26)(22,30)(24,31);(3,13)(4,17)(7,21)(8,23)(11,22)(12,27)(14,15)(16,31)(18,32)(20,30)(25,26)(28,29)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72211144, #72211163, #72228233, … |
| maximal normal subgroup of | #72211144, #72211163, #72228233, … |
| maximal quotient of | #72225651, #72317015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95258);