Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,17,21)(15,18,22,26)(16,19,24,20)(23,25,28,27);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(2,4)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(15,22)(18,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250261, #72257002, #72257004, … |
| maximal normal subgroup of | #72250261, #72257002, #72257004, … |
| maximal quotient of | #72305704, #72305720, #72308693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536294);