Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,25,21,23,28,22)(19,26,27,30,29,24,31,32);(5,7,8,6)(9,11,12,16)(10,13,14,15)(18,22)(19,27)(20,28)(23,25)(29,31)(30,32);(6,7)(10,14)(13,15)(18,23)(22,25)(24,26)(30,32);(2,4)(6,7)(11,16)(13,15)(17,19,28,31,21,29,20,27)(18,24,22,30,23,26,25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=fi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264567, #72264589, #72264591, … |
| maximal normal subgroup of | #72264567, #72264589, #72264591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343857);