Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,7)(9,10)(11,12)(15,23)(16,17)(19,27)(20,21)(24,28)(25,26);(5,6)(7,8)(13,14,18,22)(15,24,26,27)(16,20,17,21)(19,23,28,25);(2,4)(5,7,8,6)(13,15)(14,19)(16,25)(17,23)(18,26)(20,27)(21,24)(22,28);(10,12)(13,16,18,17)(14,20,22,21)(15,25,26,23)(19,27,28,24)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264881, #72264909, #72264970, … |
| maximal normal subgroup of | #72264881, #72264909, #72264970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301720);