Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(17,18,22,23,21,26,20,25)(19,24,29,30,27,31,28,32);(2,5)(3,7)(10,14)(11,12);(9,10)(11,16)(12,15)(13,14)(18,23)(20,22)(24,30)(25,26)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,24,26,31)(20,28,22,29)(23,30,25,32);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,27)(24,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fgi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282572, #72282609, #72283006, … |
| maximal normal subgroup of | #72282572, #72282609, #72283006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397413);