Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16);(1,3,5,4)(2,6,8,7)(18,22)(19,20)(23,24);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23);(1,4,5,3)(2,6,8,7)(17,18)(19,22)(20,24)(21,23);(9,11,12,15)(10,14,13,16)(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313251, #72313612, #72313857, … |
| maximal normal subgroup of | #72313251, #72313612, #72313857, … |
| maximal quotient of | #72319925, #72319993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363939);