Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(13,14)(15,17)(16,18)(19,20)(21,22,25,26,24,28,23,27);(5,6,8,10)(7,11,12,9)(13,15,16,19)(14,17,18,20)(22,26)(23,25)(27,28);(2,4)(5,7,8,12)(6,9,10,11);(14,18)(17,20)Polycyclic presentation:
2^9|a^2=ei,b^2=gh,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264196, #72264200, #72264209, … |
| maximal normal subgroup of | #72264196, #72264200, #72264209, … |
| maximal quotient of | #72307015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318050);