Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2595 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)(7,11)(8,10)(9,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,27)(26,28);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(1,2,4,3)(7,12)(9,11)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(6,10)(9,11)(14,20)(18,22)(19,21)(26,27);(1,3,4,2)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^9|c^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256903, #72256935, #72256993, … |
| maximal normal subgroup of | #72256903, #72256935, #72256993, … |
| maximal quotient of | #72265691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717154);