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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,19)(15,18,22,27)(17,20,24,28)(21,26,23,25);(2,3)(13,15)(14,18)(16,22)(17,21)(19,27)(20,26)(23,24)(25,28);(1,2,4,3)(6,9)(10,12)(14,19)(15,21)(17,24)(18,25)(22,23)(26,27);(1,3,4,2)(15,21)(17,24)(18,26)(20,28)(22,23)(25,27);(5,7,8,11)(6,10,9,12)(15,22)(18,27)(21,23)(25,26)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72257316, #72263878, … |
| maximal normal subgroup of | #72239609, #72257316, #72263878, … |
| maximal quotient of | #72239728, #72264522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363572);