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 | #44915 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,27,26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(16,23)(17,25)(19,27)(20,22)(21,28)(24,26);(2,4)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27);(6,11)(10,12)(14,15)(16,24)(20,22)(23,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72263968, #72263976, #72264061 |
| maximal normal subgroup of | #72263968, #72263976, #72264061 |
| maximal quotient of | #72264257 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424774);