Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #44871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(18,23)(19,27)(20,21)(24,29)(25,26)(28,30);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,21)(25,26)(27,30)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,18,20,23,22,26,21,25)(19,29,27,32,30,24,28,31);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,19,21,28,22,30,20,27)(18,24,25,32,26,29,23,31);(2,6)(7,8)(10,14)(13,15)(19,30)(24,29)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=gh,c^2=fg,d^2=fgi,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^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257454, #72257568, #72264193 |
| maximal normal subgroup of | #72257454, #72257568, #72264193 |
| maximal quotient of | #72307868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499536);