Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| 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 | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,16,20,18,21,17,22)(15,23,24,28,26,27,25,19);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,19,21,28)(16,25,17,24)(20,27,22,23);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(5,7,8,11)(6,10,9,12);(14,21)(19,28)(20,22)(23,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264217, #72264288, #72264299, … |
| maximal normal subgroup of | #72264217, #72264288, #72264299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339803);