Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,19,24,27)(16,22,18,26)(21,25,28,23);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,27)(21,26)(22,28);(6,10)(9,12)(14,20)(19,27)(21,28)(22,26);(1,3,4,2)(13,15)(14,21)(16,23)(17,24)(18,25)(19,26)(20,28)(22,27);(13,16,17,18)(14,22,20,26)(15,23,24,25)(19,21,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=ei,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72310659, #72310674, #72310682, … |
| maximal normal subgroup of | #72310659, #72310674, #72310682, … |
| maximal quotient of | #72313003, #72313703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536884);