Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,28,32)(23,35,39,47)(24,40,42,49)(25,31,41,48)(26,34,27,33)(29,38,46,37)(30,43,50,36)(44,52,45,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,37,28,39,27,38)(22,29,33,35,32,46,34,47)(24,41,43,45,42,25,36,44)(30,48,49,52,50,31,40,51);(6,11)(7,13)(12,15)(14,19)(21,24,26,43,28,42,27,36)(22,30,33,49,32,50,34,40)(23,25,37,44,39,41,38,45)(29,31,35,51,46,48,47,52);(7,13)(10,17)(14,19)(18,20)(21,25)(22,31)(23,36)(24,37)(26,44)(27,45)(28,41)(29,40)(30,35)(32,48)(33,51)(34,52)(38,42)(39,43)(46,49)(47,50);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,32)(24,42)(25,41)(29,46)(33,34)(35,47)(36,43)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406134);