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 | |
2436 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)(15,19)(16,18)(17,21)(20,25)(22,28)(23,26)(24,27);(2,4)(14,18)(15,22)(16,25)(19,27)(20,21)(23,24);(5,7,8,11)(6,10,9,12)(13,15,17,24)(14,19,21,27)(16,23,25,22)(18,26,20,28);(14,18)(15,23)(16,25)(20,21)(22,24)(26,28)Polycyclic presentation:
2^9|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 | #72256829, #72256888, #72256891, … |
| maximal normal subgroup of | #72256829, #72256888, #72256891, … |
| maximal quotient of | #72264376, #72264482, #72264686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482354);