Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,19,22)(15,20,24,28)(16,17,21,18)(23,25,26,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,24)(14,20,22,28)(16,26,21,23)(17,27,18,25);(2,4)(13,16,18,14,19,21,17,22)(15,23,25,28,24,26,27,20);(2,4)(5,8,9,7)(6,10,11,12)(15,24)(20,28)(23,26)(25,27);(13,17,19,18)(14,21,22,16)(15,25,24,27)(20,26,28,23)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72284192, #72307338, #72307430, … |
| maximal normal subgroup of | #72284192, #72307338, #72307430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351104);