Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,11)(10,12);(5,7)(6,10)(8,11)(9,12)(14,18)(15,16)(19,20);(2,4)(5,8,9,10)(6,7,11,12)(13,14)(15,18)(16,20)(17,19);(2,4)(6,11)(7,12);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282887, #72283247, #72283300, … |
| maximal normal subgroup of | #72282887, #72283247, #72283300, … |
| maximal quotient of | #72320311 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408558);