Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3100 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,12,7,10)(14,19,22,21)(15,23,24,25);(1,2,3,4)(6,10)(7,8)(11,12)(13,14,16,19,18,22,17,21)(15,20,23,26,24,28,25,27);(6,11)(7,8)(10,12)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,27)(22,28);(5,7,9,8)(6,10,12,11)(13,16,18,17)(14,21,22,19)(15,23,24,25)(20,27,28,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225125, #72318402, #72318418, … |
| maximal normal subgroup of | #72225125, #72318402, #72318418, … |
| maximal quotient of | #72225109, #72285812, #72286602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100859);