Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | |
2279 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)(7,10)(8,9)(11,12)(14,19)(15,23)(16,25)(22,27);(6,9)(10,12)(13,14,17,19)(15,21,23,28)(16,26,25,20)(18,27,24,22);(2,4)(13,15)(14,20)(16,18)(17,23)(19,26)(21,27)(22,28)(24,25);(5,7,8,11)(6,10,9,12)(15,16)(18,24)(20,28)(21,26)(22,27)(23,25);(13,16)(14,21)(15,24)(17,25)(18,23)(19,28)(20,22)(26,27)Polycyclic presentation:
2^9|b^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257346, #72257467, #72264075 |
| maximal normal subgroup of | #72257346, #72257467, #72264075 |
| maximal quotient of | #72308808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482428);