Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2331 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)(7,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15,17,16)(14,18,20,19);(2,4)(14,19)(15,16)(18,20);(5,8,10,9)(6,11,7,12);(2,4)(5,9,10,8)(6,11,7,12)(14,20)(18,19)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280286, #72280317, #72280764, … |
| maximal normal subgroup of | #72280286, #72280317, #72280764, … |
| maximal quotient of | #72304939, #72305550, #72305721 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583996);