Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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 | #25734 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 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)(3,8)(4,6)(5,7)(19,25)(20,27)(22,29)(24,30)(26,31)(28,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,26,28,22)(20,24,27,30)(25,31,32,29);(9,11)(10,13)(12,16)(14,15)(17,19,27,32,21,28,20,25)(18,22,30,31,23,26,24,29);(11,16)(13,15)(18,23)(19,28)(24,30)(25,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,28)(22,26)(25,32)(29,31)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282696, #72283041, #72283205, … |
| maximal normal subgroup of | #72282696, #72283041, #72283205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415589);