Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14205 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,8,18,26)(5,9)(6,11)(7,14)(10,19)(13,27,16,30)(15,21)(17,24,32,25)(20,31,23,28)(22,29);(1,3)(2,7,11,22)(4,16,18,13)(5,14)(6,15)(8,23)(9,21,10,12)(17,31,32,28)(19,29)(20,26)(24,27)(25,30);(1,4,5,17,6,18,19,32)(2,8,10,25,11,26,9,24)(3,13,14,28,15,16,29,31)(7,20,12,27,22,23,21,30);(2,9,11,10)(7,21,22,12)(8,24,26,25)(20,30,23,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319853 |
| maximal normal subgroup of | #72319853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285243);