Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,19,16)(15,22,25,28)(17,21,18,26)(20,27,23,24);(6,9)(7,11)(13,15,17,27,19,25,18,24)(14,20,26,28,16,23,21,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,26)(18,21)(20,25)(22,27)(24,28);(6,9)(10,12)(13,16,19,14)(15,22,25,28)(17,26,18,21)(20,27,23,24);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282981, #72283123, #72283416, … |
| maximal normal subgroup of | #72282981, #72283123, #72283416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056549);