Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,17)(12,22)(13,18)(14,23)(15,19)(16,21)(20,24);(5,6)(7,8)(9,11)(10,16)(12,15)(13,21)(14,20)(17,23)(18,22)(19,24);(1,2,3,4)(6,8)(9,12,15,11)(10,17,19,22)(13,21,14,20)(16,18,24,23);(6,8)(9,13,15,14)(10,18,19,23)(11,20,12,21)(16,17,24,22);(9,13,15,14)(10,18,19,23)(11,21,12,20)(16,22,24,17)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342372, #72342373, #72342374, … |
| maximal normal subgroup of | #72342372, #72342373, #72342374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349728);