Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33519 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,17,26,16)(7,13,20,31,21,33,36,32)(8,14,22,12,24,28,9,25)(18,29,19,34,23,30,35,27);(6,12)(8,22)(9,24)(10,26)(13,27)(14,15)(16,28)(17,25)(18,19)(20,36)(23,35)(29,32)(30,31)(33,34);(2,4)(5,7,11,21)(6,13,17,33)(8,23,24,18)(9,19,22,35)(10,20,26,36)(12,27,25,34)(14,30,28,29)(15,31,16,32);(5,8,10,22,11,24,26,9)(6,14,15,12,17,28,16,25)(7,18,20,19,21,23,36,35)(13,29,31,34,33,30,32,27)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231551, #72249766, #72250867, … |
| maximal normal subgroup of | #72231551, #72249766, #72250867, … |
| maximal quotient of | #72250110, #72257073, C2×#495730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402926);