Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #24372 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2092 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)(5,6,8,9)(7,10,11,12)(14,19)(15,20)(21,26)(23,27);(6,9)(10,12)(13,14)(15,22)(16,23)(17,19)(18,26)(20,28)(21,25)(24,27);(2,4)(13,15,17,20)(14,16,19,24)(18,27,25,23)(21,28,26,22);(5,7,8,11)(6,10,9,12)(15,23)(16,22)(18,25)(20,27)(21,26)(24,28);(13,16)(14,20)(15,19)(17,24)(18,28)(21,23)(22,25)(26,27)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263905, #72263935, #72263990 |
| maximal normal subgroup of | #72263905, #72263935, #72263990 |
| maximal quotient of | #72264214, #72265186 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483388);