Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2399 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);(6,9)(10,12)(13,14,18,16)(15,22,24,28)(17,20,27,21)(19,26,25,23);(2,4)(13,15,18,24)(14,19,16,25)(17,26,27,23)(20,22,21,28);(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,22,24,28)(17,21,27,20)(19,26,25,23);(14,20)(16,21)(17,27)(19,22)(23,26)(25,28)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251279, #72256879, #72256896, … |
| maximal normal subgroup of | #72251279, #72256879, #72256896, … |
| maximal quotient of | #72264240, #72264559, #72280820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482362);