Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | #36152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,21)(17,23)(19,25)(20,27)(22,24)(26,28);(2,4)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(5,8,9,7)(6,10,11,12)(13,15)(14,18)(16,21)(17,23)(19,25)(20,27)(22,24)(26,28);(15,21)(16,24)(18,25)(19,28)(22,23)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238725, #72249378, #72249430, … |
| maximal normal subgroup of | #72238725, #72249378, #72249430, … |
| maximal quotient of | #72239773, #72264246, #72307812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483746);