Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3292 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,5,22)(3,14,16,9,18,27,17,28)(4,10,19,30,21,31,20,29)(6,24,15,32,25,26,23,12);(1,3)(2,9)(4,6)(5,17)(7,16)(8,18)(10,12)(11,27)(13,28)(14,22)(15,19)(20,23)(21,25)(24,30)(26,29)(31,32);(1,4)(2,9)(3,6)(5,19)(7,20)(8,21)(10,26)(11,14)(12,29)(13,28)(15,17)(16,23)(18,25)(22,27)(24,31)(30,32);(1,4)(2,10)(3,15)(5,20)(6,17)(7,19)(8,21)(9,26)(11,30)(12,27)(13,31)(14,32)(16,25)(18,23)(22,29)(24,28)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fhi,g^c=gi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214153, #72214250, … |
| maximal normal subgroup of | #72214039, #72214153, #72214250, … |
| maximal quotient of | #72216438, #72227030, #72227033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404606);