Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(14,18)(15,22)(16,20)(21,26)(23,24)(25,27);(1,3,4,2)(5,6,8,10)(7,11,12,9)(13,14,16,18,17,21,20,26)(15,19,23,25,24,28,22,27);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,22)(21,28)(26,27);(13,14,20,26,17,21,16,18)(15,19,22,27,24,28,23,25)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=g,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321477, #72321592, #72321621, … |
| maximal normal subgroup of | #72321477, #72321592, #72321621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277266);