Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4304 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)(3,6)(4,7)(5,8)(12,20)(14,17)(16,22)(18,24);(3,4)(6,7)(9,10)(11,17)(12,16)(13,15)(14,19)(18,23)(20,22)(21,24);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,15,17)(12,18,20,24)(16,21,22,23);(1,4,5,3)(2,7,8,6)(10,15)(12,20)(14,17)(18,24);(2,8)(6,7)(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252416, #72252426, #72306475 |
| maximal normal subgroup of | #72252416, #72252426, #72306475 |
| maximal quotient of | #72267155 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394855);