Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,26)(22,32)(24,34)(27,35)(28,31)(29,33)(30,39)(36,38)(37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,32)(18,34)(19,21)(20,31)(25,40)(26,38)(27,39)(29,37)(30,33)(35,36);(9,12,14,26,15,27,13,29)(10,17,19,25,20,33,18,35)(11,22,23,38,24,39,28,37)(16,30,21,36,31,32,34,40);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307003, #72307337, #72307382 |
| maximal normal subgroup of | #72307003, #72307337, #72307382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313381);