Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2322 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)(13,14,18,22)(15,19,25,28)(16,20,17,21)(23,26,24,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,23)(18,25)(20,27)(21,26)(22,28);(2,4);(2,4)(5,7,8,11)(6,10,9,12)(15,23)(16,17)(19,26)(20,21)(24,25)(27,28);(6,9)(10,12)(13,16,18,17)(14,20,22,21)(15,24,25,23)(19,27,28,26)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72310818, #72310820, #72310860, … |
| maximal normal subgroup of | #72310818, #72310820, #72310860, … |
| maximal quotient of | #72311783, #72311982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423233);