Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| 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 | |
1726 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)(24,35)(25,29)(26,40)(27,33)(28,38)(30,39)(32,46)(34,42)(36,50)(37,49)(41,51)(43,47)(44,52)(45,48);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23)(22,27)(24,36)(25,33)(26,34)(28,43)(29,31)(30,42)(32,37)(35,46)(38,51)(39,48)(40,45)(41,44)(47,52)(49,50);(6,11)(7,13)(12,15)(14,19)(21,24,26,38,25,37,39,52)(22,28,30,35,29,44,40,49)(23,32,34,47,33,36,48,51)(27,41,42,50,31,43,45,46);(7,13)(10,17)(14,19)(18,20)(23,33)(24,37)(27,31)(28,44)(34,48)(35,49)(38,52)(42,45);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,29)(23,33)(28,44)(30,40)(32,36)(34,48)(35,49)(47,51)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,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=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405739);