Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3580 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)(7,10)(8,9)(11,12)(15,21)(16,23)(18,25)(20,27)(22,24)(26,28);(1,2)(3,4)(6,9)(10,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,26)(23,27)(24,28);(2,4)(13,15,17,22)(14,18,19,26)(16,24,23,21)(20,28,27,25);(5,7,8,11)(6,10,9,12);(14,19)(18,26)(20,27)(25,28)Polycyclic presentation:
2^9|c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307380, #72307439, #72307586, … |
| maximal normal subgroup of | #72307380, #72307439, #72307586, … |
| maximal quotient of | #72315978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339235);