Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #28467 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,10)(7,11,12,9)(14,18)(15,21)(20,26)(22,24);(1,2,4,3)(7,12)(9,11)(14,18)(15,22)(17,23)(19,27)(21,24)(25,28);(5,7,8,12)(6,9,10,11)(13,14,16,18)(15,19,21,25)(17,20,23,26)(22,28,24,27);(6,10)(9,11)(13,15)(14,19)(16,21)(17,24)(18,25)(20,27)(22,23)(26,28);(1,3,4,2)(14,18)(19,25)(20,26)(27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72263913, #72263918, #72263965 |
| maximal normal subgroup of | #72263913, #72263918, #72263965 |
| maximal quotient of | #72264168, #72264385 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7445257);