Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391960 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14666 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,7,13,24)(4,14)(5,18)(6,9)(8,16)(10,20)(11,28,15,32)(12,22,31,23)(17,27)(19,26)(21,30,25,29);(1,3)(2,7,9,24)(4,15,17,11)(5,12)(6,13)(8,25)(10,23,18,22)(14,32)(16,30,19,29)(20,31)(21,26)(27,28);(1,4)(2,8)(3,11)(5,19)(6,17)(7,21)(9,26)(10,27)(12,30)(13,15)(14,18)(16,20)(22,28)(23,32)(24,25)(29,31);(3,12)(5,20)(7,22)(8,26)(10,18)(11,29)(13,31)(15,30)(16,19)(21,28)(23,24)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251777, #72251785, #72251803, … |
| maximal normal subgroup of | #72251777, #72251785, #72251803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299156);