Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,21)(15,22,25,28)(16,18,26,17)(20,23,27,24);(1,2,3,4)(7,8)(10,12)(13,15,18,24,19,25,17,23)(14,20,16,22,21,27,26,28);(5,7,9,8)(6,10,11,12)(13,16,18,21,19,26,17,14)(15,22,24,27,25,28,23,20);(5,8,9,7)(6,10,11,12);(13,17,19,18)(14,16,21,26)(15,23,25,24)(20,22,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279751, #72279858, #72280255, … |
| maximal normal subgroup of | #72279751, #72279858, #72280255, … |
| maximal quotient of | #72313037, #72313038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370381);