Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2420 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)(7,11)(8,10)(9,12)(14,18)(15,16)(19,20);(1,2)(3,4)(7,12)(9,11)(13,14,15,19,17,20,16,18);(5,7,8,12)(6,9,10,11)(13,15,17,16)(14,18,20,19);(2,4)(6,10)(9,11)(13,15,17,16)(14,19,20,18);(13,14,15,19,17,20,16,18)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=fi,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=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257098, #72257122, #72257196, … |
| maximal normal subgroup of | #72257098, #72257122, #72257196, … |
| maximal quotient of | #72264739, #72264863, #72308781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7440873);