Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | |
2752 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)(21,22,25,29,27,32,26,31)(23,33,35,44,37,42,36,28)(24,38,40,46,39,30,41,45)(34,48,50,43,49,47,51,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,37)(22,28,32,44)(24,34,39,49)(25,36,26,35)(29,42,31,33)(30,43,38,52)(40,51,41,50)(45,48,46,47);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,29)(25,26)(28,42)(30,45)(31,32)(33,44)(35,36)(38,46)(40,41)(43,48)(47,52)(50,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,30)(23,34)(25,40)(26,41)(27,39)(28,43)(29,45)(31,46)(32,38)(33,47)(35,50)(36,51)(37,49)(42,48)(44,52);(6,12)(10,13)(11,20)(18,19)(24,39)(30,38)(34,49)(40,41)(43,52)(45,46)(47,48)(50,51)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436886);