Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,10)(9,22)(13,28)(14,16)(15,23)(18,30)(19,34)(20,24)(21,35)(25,26)(29,33)(31,36);(1,2)(3,4)(5,6,8,12,11,14,10,16)(7,18,20,13,21,31,19,29)(9,23,22,32,26,15,25,27)(17,33,34,36,35,28,24,30);(2,4)(5,7,8,19,11,21,10,20)(6,13,12,18,14,29,16,31)(9,24,22,35,26,34,25,17)(15,28,27,36,23,33,32,30);(6,14)(12,16)(13,29)(15,23)(18,31)(27,32)(28,33)(30,36)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231495, #72231523, #72231555, … |
| maximal normal subgroup of | #72231495, #72231523, #72231555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406250);