Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,16,19,21,17,25)(15,20,23,22,24,28,26,27);(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,25)(19,21)(22,26)(23,27)(24,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,22)(17,26)(18,23)(19,24)(21,28)(25,27);(7,12)(9,11)(13,16,17,14,19,25,18,21)(15,22,26,20,24,27,23,28);(6,10)(9,11)(13,17,19,18)(14,16,21,25)(15,23,24,26)(20,27,28,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257042, #72257061, #72257078, … |
| maximal normal subgroup of | #72257042, #72257061, #72257078, … |
| maximal quotient of | #72264718, #72264741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928814);