Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
399 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)(19,29,31,40)(20,33)(21,25)(22,37)(23,28)(24,30,41,47)(26,34)(27,39)(32,44,38,46)(35,45,48,42)(36,43);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19)(18,24)(20,30)(21,35)(22,38)(23,31)(25,40)(26,44)(27,45)(28,41)(29,43)(32,39)(33,46)(34,47)(36,48)(37,42);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,22,21,23,34,39,36)(18,25,27,33,28,43,37,26)(19,30,38,35,31,47,32,48)(24,40,45,46,41,29,42,44);(3,11)(4,13)(7,10)(12,15)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216701, #72216779, #72216788, … |
| maximal normal subgroup of | #72216701, #72216779, #72216788, … |
| maximal quotient of | C2×#484139, #72318512, #72318708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391335);