Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34977 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,25,22,26,21,20)(19,27,30,24,32,28,31,29);(2,6)(7,8)(17,19)(18,24)(20,28)(21,30)(22,32)(23,31)(25,27)(26,29);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,18,23,25,22,26,21,20)(19,28,30,29,32,27,31,24);(1,4,5,3)(2,8,6,7)(17,20,21,26,22,25,23,18)(19,29,31,28,32,24,30,27);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,20)(19,30,32,31)(24,27,29,28)Polycyclic presentation:
2^9|a^2=hi,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256848, #72256864, #72256968, … |
| maximal normal subgroup of | #72256848, #72256864, #72256968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366422);