Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #2215 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(17,18)(19,28)(20,26)(21,33)(22,37)(23,25)(24,30)(27,35)(29,46)(31,45)(32,42)(34,40)(36,44)(38,43)(39,47)(41,48);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19,23,31)(18,24,25,39)(20,29,22,38)(21,34,36,48)(26,40,37,41)(27,43,42,46)(28,44,45,33)(30,32,47,35);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(18,25)(24,39)(26,37)(27,42)(28,45)(33,44)(40,41)(43,46);(3,11)(4,13)(7,10)(12,15)(17,20,23,22)(18,26,25,37)(19,29,31,38)(21,32,36,35)(24,40,39,41)(27,33,42,44)(28,46,45,43)(30,34,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234336, #72234347, #72234348, … |
| maximal normal subgroup of | #72234336, #72234347, #72234348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391316);