Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,8)(9,10,13,14)(11,12,15,16)(25,26)(27,31)(28,29)(30,32);(2,6)(3,8)(10,14)(11,15)(18,21)(22,24)(26,29)(30,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,15)(14,16);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,16)(14,15)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=gi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238503, #72238508, #72263212 |
| maximal normal subgroup of | #72238503, #72238508, #72263212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 785787);