Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,33)(24,28)(25,30)(26,40)(27,32)(29,35)(31,41)(34,47)(36,42)(37,43)(38,45)(39,51)(44,48)(46,50)(49,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,24,27,35,41,37)(22,28,31,33,32,43,40,29)(25,34,39,36,38,48,50,49)(30,42,46,47,45,52,51,44);(7,13)(8,15)(10,16)(11,18)(21,24,26,35,27,37,41,23)(22,29,31,28,32,33,40,43)(25,36,39,48,38,49,50,34)(30,44,46,42,45,47,51,52);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25)(22,30)(23,34)(24,36)(26,39)(27,38)(28,42)(29,44)(31,46)(32,45)(33,47)(35,48)(37,49)(40,51)(41,50)(43,52);(7,13)(10,16)(14,19)(17,20)(25,38)(30,45)(34,48)(36,49)(39,50)(42,52)(44,47)(46,51)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434264);