Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(9,10,11,12)(14,17)(18,20)(21,22,25,23)(24,27,28,26);(1,2,3,4)(6,8)(14,17)(15,19)(22,23)(24,28);(1,2)(3,4)(5,6)(7,8)(13,14)(15,18)(16,17)(19,20)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72267504, #72267505, #72267738, … |
| maximal normal subgroup of | #72267504, #72267505, #72267738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 63703);