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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(14,19)(20,26)(21,22)(27,28);(2,3)(6,9)(10,12)(13,14)(15,20)(16,22)(17,19)(18,21)(23,26)(24,28)(25,27);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,15,17,23)(14,20,19,26)(16,25,18,24)(21,28,22,27);(1,3,4,2)(14,21)(15,23)(16,18)(19,22)(20,27)(26,28);(6,9)(10,12)(13,16,17,18)(14,22,19,21)(15,24,23,25)(20,28,26,27)Polycyclic presentation:
2^9|c^2=fg,d^2=g,e^2=i,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 | #72307820, #72307916, #72308520 |
| maximal normal subgroup of | #72307820, #72307916, #72308520 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467844);