Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(14,17)(18,20)(21,22,25,26)(23,24,28,27);(6,9)(10,12)(15,19)(18,20)(22,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,28)(22,27,26,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72293912, C2×#606717, #72336965, … |
| maximal normal subgroup of | #72293912, C2×#606717, #72336965, … |
| maximal quotient of | #72293960, C2×#606717 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 513913);