Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,20,16,17);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,14,18,16,20,19);(6,10)(7,11)(14,19)(15,16);(2,4)(6,10)(9,12)(14,19)(15,16);(13,16,17,19,18,15,20,14)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257451, #72257505, #72257767, … |
| maximal normal subgroup of | #72257451, #72257505, #72257767, … |
| maximal quotient of | #72264730, #72307861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434755);