Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31);(2,6)(7,8)(17,19)(18,24)(20,29)(21,25)(22,28)(23,30)(26,32)(27,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,28)(20,31)(21,22)(24,32)(25,30)(26,29);(9,11,12,15)(10,14,13,16)(17,20,23,32)(18,25,27,28)(19,29,30,26)(21,31,22,24);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(22,23)(24,31)(25,28)(27,29)Polycyclic presentation:
2^9|c^2=f,d^2=gi,e^2=f,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 | #72284983, #72285025 |
| maximal normal subgroup of | #72284983, #72285025 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6495014);