Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,12,16,14,18,13,17)(11,19,22,15,21,23,20,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,23)(18,24);(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,23)(18,24);(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,24,23);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,23,24,19)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282030, #72282085, #72282387, … |
| maximal normal subgroup of | #72282030, #72282085, #72282387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345045);