Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| 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 | |
1744 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)(23,32)(24,36)(25,30)(26,31)(27,42)(28,35)(29,40)(33,48)(34,45)(37,51)(38,47)(39,50)(41,44)(43,52)(46,49);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23)(22,28)(24,37)(25,41)(26,35)(27,34)(29,46)(30,45)(31,32)(33,39)(36,48)(38,49)(40,52)(42,44)(43,47)(50,51);(6,11)(7,13)(12,15)(14,19)(21,24,27,40,26,39,25,38)(22,29,30,36,31,47,42,50)(23,33,34,49,35,37,41,52)(28,43,45,51,32,46,44,48);(7,13)(10,17)(14,19)(18,20)(21,25,26,27)(22,30,31,42)(23,34,35,41)(24,38,39,40)(28,44,32,45)(29,36,47,50)(33,49,37,52)(43,48,46,51);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(23,35)(29,47)(30,42)(33,37)(34,41)(36,50)(49,52)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405734);