Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497838 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13,16,29)(5,20)(6,21)(9,24,26,31)(10,22)(11,12)(14,27,30,23)(15,25,32,28)(17,18);(1,2)(3,9)(4,6)(5,11)(7,12)(8,20)(10,17)(13,14)(15,27)(16,26)(18,21)(19,22)(23,32)(24,25)(28,31)(29,30);(1,3,6,13,5,15,21,28,7,16,22,29,20,32,10,25)(2,9,4,14,11,27,18,31,12,26,19,30,8,23,17,24);(1,4)(2,10)(3,14)(5,18)(6,11)(7,19)(8,22)(9,25)(12,21)(13,27)(15,31)(16,30)(17,20)(23,29)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214178, #72214308, … |
| maximal normal subgroup of | #72214039, #72214178, #72214308, … |
| maximal quotient of | #72216392, #72227118, #72227119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405034);