Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360529 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13877 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,8,5,20)(3,12,16,22,13,27,15,9)(4,17,18,24,19,10,21,25)(14,29,30,32,28,26,31,23);(2,8)(3,13)(5,6)(9,22)(10,17)(14,28)(18,21)(23,32);(1,3,5,15,7,13,6,16)(2,9,11,12,8,22,20,27)(4,14,18,30,19,28,21,31)(10,23,24,26,17,32,25,29);(1,4)(2,10)(3,14)(5,18)(6,21)(7,19)(8,17)(9,23)(11,24)(12,26)(13,28)(15,30)(16,31)(20,25)(22,32)(27,29)Polycyclic presentation:
2^9|a^2=h,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72241303, #72241304, #72241305, … |
| maximal normal subgroup of | #72241303, #72241304, #72241305, … |
| maximal quotient of | C2×#360531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267727);