Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,21,17,20,24,18)(15,19,22,27,23,26,28,25);(5,7,9,8)(6,10,11,12)(14,18)(15,22)(16,24)(20,21)(23,28)(25,27);(5,8,9,7)(6,10,11,12)(15,23)(19,26)(22,28)(25,27);(13,15,16,22,17,23,24,28)(14,19,21,27,20,26,18,25);(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239749, #72249399, #72249419, … |
| maximal normal subgroup of | #72239749, #72249399, #72249419, … |
| maximal quotient of | #72309021, C2×#6380656, #72313690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287852);