Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C22×#447 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,16,19,17,20,15,18);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,20);(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250411, #72250415, #72250438, … |
| maximal normal subgroup of | #72250411, #72250415, #72250438, … |
| maximal quotient of | #72250879, #72250965, #72250968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102398);