Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417538 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,13,18,22)(12,19,21,15)(14,16)(17,23)(20,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13,16,22)(12,17)(14,18)(15,20,19,23)(21,24);(3,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(9,13)(10,11)(12,20)(14,22)(15,17)(16,18)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240905, #72266049, #72266050, … |
| maximal normal subgroup of | #72240905, #72266049, #72266050, … |
| maximal quotient of | #72240909, #72303387 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358324);