Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1782 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C2 | C2×#1782 |
| prime radical | #62814 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,9)(5,17,18,7)(6,10)(8,14)(12,24)(13,19,27,23)(15,28,29,20)(16,22)(21,26)(25,30,32,31);(2,7)(8,19)(9,20)(10,17)(11,23)(21,30)(22,28)(24,31);(1,3)(2,8,10,11)(4,12)(5,13)(6,14)(7,19,17,23)(9,21,22,24)(15,25)(16,26)(18,27)(20,30,28,31)(29,32);(1,4,6,16)(2,9,10,22)(3,12,14,26)(5,15,18,29)(7,20,17,28)(8,21,11,24)(13,25,27,32)(19,30,23,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,f^c=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285444, #72303558, #72303561, … |
| maximal normal subgroup of | #72285444, #72303558, #72303561, … |
| maximal quotient of | #72284804, #72303122, #72303416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400653);