Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| 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 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2464 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,25)(17,20)(18,21)(22,26)(23,27)(24,28);(6,9)(10,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,22)(21,27)(25,28);(2,4)(13,14)(15,22)(16,20)(17,25)(18,21)(19,26)(23,28)(24,27);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(22,28)(24,26);(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,22,27,28)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250065, #72250181, #72256994, … |
| maximal normal subgroup of | #72250065, #72250181, #72256994, … |
| maximal quotient of | #72308459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339479);