Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 7340032 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1278 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 490 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(11,17)(12,19)(14,20)(15,22)(26,30)(27,33)(28,35)(38,40)(43,49)(47,52)(50,55)(54,56);(1,3,4,8)(2,5,6,7)(11,17)(14,20)(18,23)(21,24)(28,35)(32,37)(34,39)(38,40)(42,46)(43,49)(48,53)(50,55);(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40)(41,42)(43,47)(44,48)(45,46)(49,52)(50,54)(51,53)(55,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40)(41,43,45,49)(42,47,46,52)(44,50,51,55)(48,54,53,56);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)(41,44,45,51)(42,48,46,53)(43,50,49,55)(47,54,52,56)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039829);