Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #24434 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,23,25,19)(16,21,18,20)(24,27,26,28);(1,2)(3,4)(6,9)(10,12)(13,15,17,25)(14,19,22,23)(16,24,18,26)(20,27,21,28);(2,4)(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(2,4);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,28,23,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72256938, #72256952, … |
| maximal normal subgroup of | #72239609, #72256938, #72256952, … |
| maximal quotient of | #72239673, #72264369, #72264442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347068);