Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,31)(21,27)(22,26)(23,28)(24,29)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,29)(20,30)(21,22)(24,25)(26,27)(31,32);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,19,21,29,23,30,22,20)(18,24,26,25,28,31,27,32);(1,3,5,4)(2,7,8,6)(11,16)(14,15)(17,20,21,19,23,29,22,30)(18,25,26,31,28,32,27,24);(1,4,5,3)(2,7,8,6)(19,30)(20,29)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fhi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72340627, #72340641, #72340653 |
| maximal normal subgroup of | #72340627, #72340641, #72340653 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7512946);