Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 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)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,31)(24,29)(25,27)(26,38)(28,35)(30,32)(33,44)(34,43)(36,40)(37,49)(39,45)(41,48)(42,46)(47,51)(50,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(22,27)(28,39)(29,40)(30,38)(31,43)(41,50)(42,49)(44,51);(6,11)(7,13)(12,15)(14,19)(21,23,32,45,25,34,26,35)(22,28,38,43,27,39,30,31)(24,33,46,52,36,47,37,48)(29,41,49,51,40,50,42,44);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,33)(25,36)(26,37)(27,40)(28,41)(30,42)(31,44)(32,46)(34,47)(35,48)(38,49)(39,50)(43,51)(45,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,36)(29,40)(33,47)(37,46)(41,50)(42,49)(44,51)(48,52)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405762);