Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4130 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)(5,6)(7,10)(8,9)(11,12)(14,18)(16,23)(19,25)(20,24)(22,27)(26,28);(6,9)(10,12);(13,14,17,20)(15,19,21,26)(16,24,23,18)(22,28,27,25);(2,4)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(5,7,8,11)(6,10,9,12)(15,21)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72315644, #72315715, #72315732 |
| maximal normal subgroup of | #72315644, #72315715, #72315732 |
| maximal quotient of | #72341956, #72341963, #72341971 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536494);