Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C2×#3264 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2245 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,21)(18,23)(19,22)(20,26)(24,27)(25,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,17)(16,24)(18,20)(19,25)(21,27)(22,28)(23,26);(13,16)(17,27)(18,19)(20,24)(21,23)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,26)(18,27)(19,28)(21,25)(22,24);(6,9)(10,12)(13,16,19,18)(14,21,22,23)(15,20,25,24)(17,26,28,27)Polycyclic presentation:
2^9|d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72263905, #72263918, #72263925, … |
| maximal normal subgroup of | #72263905, #72263918, #72263925, … |
| maximal quotient of | #72264326, #72264775 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424209);