Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(13,14,18,17,19,22,16,21)(15,23,24,25,26,27,28,20);(9,10,11,12)(13,15)(14,20)(16,24)(17,27)(18,28)(19,26)(21,23)(22,25);(2,4)(13,16,19,18)(14,21,22,17)(15,24,26,28)(20,23,25,27);(13,17,18,22,19,21,16,14)(15,25,24,27,26,20,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72265171, #72265213, #72265316, … |
| maximal normal subgroup of | #72265171, #72265213, #72265316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263419);