Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
18352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(2,4);(6,8);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72212205, #72212847, #72227567, … |
| maximal normal subgroup of | #72212205, #72258278, #72259221, … |
| maximal quotient of | C2×#6628846, #72267484, #72268065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536042);