Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,14)(8,22)(9,24)(10,16)(11,18)(12,17)(13,20)(15,26)(19,33)(21,29)(23,28)(25,32)(27,31)(30,35)(34,36);(1,2,4,3)(5,7)(6,13)(8,12)(9,19)(10,20)(11,21)(14,16)(15,28)(17,29)(18,22)(23,27)(24,36)(25,34)(26,35)(30,31)(32,33);(5,8,11,7,10,21,12,20)(6,14,17,22,16,13,18,29)(9,23,26,19,25,35,27,34)(15,30,24,33,31,28,32,36);(1,3,4,2)(5,9,10,25)(6,15,16,31)(7,19,20,34)(8,23,21,35)(11,26,12,27)(13,28,14,30)(17,24,18,32)(22,33,29,36)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305564);