Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10)(8,17)(9,23)(11,15)(12,16)(13,21)(14,26)(18,25)(19,32)(20,29)(22,28)(24,30)(27,33)(31,35)(34,36);(5,7,11,16,12,15,10,6)(8,13,21,17,22,29,20,28)(9,18,25,23,26,30,24,14)(19,27,34,31,35,36,33,32);(1,2,3,4)(5,8,11,21,12,22,10,20)(6,13,7,17,16,29,15,28)(9,19,25,34,26,35,24,33)(14,27,18,31,23,36,30,32);(5,9,10,24,12,26,11,25)(6,14,15,30,16,23,7,18)(8,19,20,33,22,35,21,34)(13,27,28,32,29,36,17,31)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 305648);