Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,18,22,19,23,17,16)(15,21,25,20,27,28,26,24);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,28)(17,25)(18,26)(19,27)(21,22)(23,24);(5,8,9,7)(6,10,11,12)(13,15)(14,21)(16,24)(17,26)(18,25)(19,27)(20,22)(23,28);(13,16,18,14,19,22,17,23)(15,24,25,21,27,20,26,28);(13,17,19,18)(14,16,23,22)(15,25,27,26)(20,28,24,21)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72250423, #72250650, #72256854 |
| maximal normal subgroup of | #72250423, #72250650, #72256854 |
| maximal quotient of | #72308250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287864);