Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(10,11)(14,17)(18,20)(21,22,24,26)(23,27,28,25);(2,4)(5,6,8,7)(10,11)(14,17)(15,19)(23,28)(25,27);(6,7)(9,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=efgh,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72330593, #72330626, #72330642, … |
| maximal normal subgroup of | #72330593, #72330626, #72330642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67308);