Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,15,9,17,26,35,12,16,29,34,10,14,22,31)(7,18,13,30,19,8,21,24,20,23,32,36,27,25,33,28);(2,4)(5,7)(6,8,16,25)(9,19)(10,27)(11,13)(12,20)(14,18,17,23)(15,24,34,28)(21,26)(22,33)(29,32)(30,35,36,31);(5,8,22,36,10,18,29,24,12,25,26,30,9,23,11,28)(6,13,31,20,14,33,34,19,16,32,35,7,17,21,15,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279984, #72280200, #72280229 |
| maximal normal subgroup of | #72279984, #72280200, #72280229 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54577);