Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2764 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)(5,6,8,7)(9,10,14,17,13,15,22,19)(11,12,16,21,18,20,23,24);(6,7)(9,11,14,16,13,18,22,23)(10,12,17,21,15,20,19,24);(2,4)(10,15)(11,18)(16,23)(17,19);(5,7,8,6)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,18)(22,24);(10,15)(12,20)(17,19)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=dfi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72312949, #72312975, #72313006 |
| maximal normal subgroup of | #72312949, #72312975, #72313006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289021);