Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2212 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4)(6,9)(10,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(2,4)(6,10)(8,12)(9,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27);(6,9)(10,11)(14,20)(18,21)(19,26)(25,27);(2,4)(15,23)(18,21)(22,28)(25,27)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280764, #72280879, #72281990, … |
| maximal normal subgroup of | #72280764, #72280879, #72281990, … |
| maximal quotient of | #72305070, #72305798, #72305809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576848);