Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(13,14,19,16)(15,22,18,20)(17,21,26,24)(23,28,25,27);(10,12)(13,15,19,18)(14,20,16,22)(17,23,26,25)(21,27,24,28);(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,16,19,14)(15,22,18,20)(17,21,26,24)(23,27,25,28);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(13,18,19,15)(14,22,16,20)(17,25,26,23)(21,28,24,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72267451, #72267455, #72267459, … |
| maximal normal subgroup of | #72267451, #72267455, #72267459, … |
| maximal quotient of | #72267485, #72311861, #72314434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424612);