Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2)(3,4)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(6,9)(7,12)(10,11)(15,21)(18,24)(22,27)(25,28);(2,4)Polycyclic presentation:
2^9|a^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,f^c=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72318330, #72318344, #72318415, … |
| maximal normal subgroup of | #72318330, #72318344, #72318415, … |
| maximal quotient of | #72318675, #72318781, #72318800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128684);