Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,22)(16,19)(17,21)(18,25)(20,23)(24,27)(26,28);(1,2)(3,4)(6,10)(11,12)(13,14)(15,18)(16,20)(17,21)(19,23)(22,25)(24,26)(27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,23,28,17,24,16,25)(14,18,19,27,21,26,20,22);(2,4)(5,8,9,7)(6,12,10,11)(15,24)(18,26)(22,27)(25,28);(13,16,17,23)(14,19,21,20)(15,25,24,28)(18,27,26,22)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250121, #72250131, #72250181, … |
| maximal normal subgroup of | #72250121, #72250131, #72250181, … |
| maximal quotient of | #72308541 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351272);