Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22,27,30)(23,33,35,44)(24,38,40,47)(25,32,26,31)(28,37,45,36)(29,42,48,41)(34,46,39,43)(49,51,52,50);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,39)(25,37)(26,36)(27,35)(29,46)(30,45)(31,44)(32,33)(34,40)(38,49)(41,50)(42,51)(43,48)(47,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,40)(22,29,30,48)(23,34,35,39)(25,41,26,42)(28,43,45,46)(31,38,32,47)(33,49,44,52)(36,50,37,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,30)(23,35)(24,40)(31,32)(36,37)(41,42)(43,46)(49,52);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,31,30,32)(23,36,35,37)(24,41,40,42)(28,33,45,44)(29,38,48,47)(34,50,39,51)(43,49,46,52)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436350);