Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8,11,12)(9,10)(13,14,16,18,17,20,15,19);(2,4)(5,7)(6,8)(9,11)(10,12)(14,18)(15,16)(19,20);(5,8)(6,7)(9,12)(10,11);(6,10)(8,12)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72240328, #72240330, #72240332, … |
| maximal normal subgroup of | #72240328, #72240330, #72240332, … |
| maximal quotient of | #72261493, C2×#185958, #72280466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93154);