Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 |
| prime radical | #25736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,19,17,21,16,22)(15,20,25,26,23,27,24,28);(2,3)(14,19)(16,18)(20,26)(21,22)(24,25)(27,28);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,27)(22,28);(1,3,4,2)(15,23)(20,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240756, #72257498, #72263913, … |
| maximal normal subgroup of | #72240756, #72257498, #72263913, … |
| maximal quotient of | #72240776, #72264509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374575);