Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232515 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2223 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)(12,17)(13,20)(14,19)(15,18)(16,23)(22,37)(24,32)(25,31)(26,40)(27,36)(28,34)(29,35)(30,39)(33,38);(5,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,31)(19,32)(20,21)(22,27)(28,39)(29,38)(30,35)(34,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,28)(10,17,18,34)(11,22,24,38)(13,27,14,29)(16,30,31,40)(19,35,20,36)(21,33,32,37)(23,26,25,39);(10,18)(16,31)(17,34)(19,20)(21,32)(30,40)(33,37)(35,36)Polycyclic presentation:
2^9|c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279295, #72279320 |
| maximal normal subgroup of | #72279295, #72279320 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139712);