Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2217 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)(3,6)(4,14,18,8)(5,13)(7,11)(9,19)(10,22)(12,16,24,17)(15,29,31,27)(20,25)(21,26)(23,30,28,32);(1,3,6,11,7,10,22,2)(4,14,16,24,18,8,17,12)(5,13,20,25,21,26,19,9)(15,28,32,29,31,23,30,27);(1,4)(2,8)(3,12)(5,15)(6,16)(7,18)(9,23)(10,24)(11,14)(13,27)(17,22)(19,30)(20,32)(21,31)(25,28)(26,29);(1,5)(2,9)(3,13)(4,15)(6,20)(7,21)(8,23)(10,26)(11,25)(12,27)(14,28)(16,32)(17,30)(18,31)(19,22)(24,29)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241172, #72241179, #72241302, … |
| maximal normal subgroup of | #72241172, #72241179, #72241302, … |
| maximal quotient of | C2×#366729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273925);