Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2291 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)(7,11)(8,9)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,27)(24,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,23,18,24,17,25)(14,19,26,28,21,27,20,22);(6,9)(10,11)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,22,27,28);(2,4)(5,7,8,12)(6,10,9,11)(15,24)(19,27)(22,28)(23,25);(13,15,16,23,18,24,17,25)(14,19,26,28,21,27,20,22)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257105, #72257120, #72257202, … |
| maximal normal subgroup of | #72257105, #72257120, #72257202, … |
| maximal quotient of | #72264739, #72264904, #72308317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458329);