Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#3264 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,24,20)(16,19,25,28)(17,26,27,21);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,16,18,24,27,25)(14,19,21,23,22,28,26,20);(6,9)(10,11);(2,4)(5,7,8,12)(6,10,9,11)(15,24)(16,25)(19,28)(20,23);(13,16,17,24,18,25,27,15)(14,20,21,19,22,23,26,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257391, #72257451, #72263878, … |
| maximal normal subgroup of | #72257391, #72257451, #72263878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458348);