Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3096 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)(5,6,8,10,9,7,12,11)(13,14)(15,19,20,17)(16,18);(1,2,3,4)(5,7)(6,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19);(6,7)(10,11)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,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=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240266, #72260979, #72261528, … |
| maximal normal subgroup of | #72240266, #72260979, #72261528, … |
| maximal quotient of | #72260983, #72279374, #72283903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102413);