Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,15,14,16,11,12)(18,22,25,23)(20,28)(24,30,31,29)(26,32);(1,3,4,7)(2,5,6,8)(10,15)(11,13)(12,16)(17,18)(19,23)(20,29)(21,25)(22,27)(24,26)(28,30)(31,32);(2,6)(5,8)(9,11,14,13)(10,12,16,15)(17,19,21,27)(18,23,25,22)(20,26,28,32)(24,30,31,29);(9,12,13,10,14,15,11,16)(17,20,27,32,21,28,19,26)(18,24,23,30,25,31,22,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=ci,d^b=de,e^b=eh,g^b=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343346, #72343352, #72343376, … |
| maximal normal subgroup of | #72343346, #72343352, #72343376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132448);