Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,30)(23,33,35,45)(24,38,40,48)(25,32,26,31)(28,37,44,36)(29,41,47,42)(34,43,39,46)(49,50,52,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,36,27,35,26,37)(22,28,31,33,30,44,32,45)(24,39,41,51,40,34,42,50)(29,46,48,52,47,43,38,49);(6,11)(7,13)(12,15)(14,19)(21,24,25,41,27,40,26,42)(22,29,31,48,30,47,32,38)(23,34,36,50,35,39,37,51)(28,43,33,49,44,46,45,52);(7,13)(10,17)(14,19)(18,20)(23,35)(24,40)(28,44)(29,47)(33,45)(36,37)(38,48)(41,42);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,30)(24,40)(28,44)(31,32)(33,45)(34,39)(41,42)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406076);