Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #52966 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(4,6)(5,8)(17,18,21,23)(19,26,25,22)(20,24,28,30)(27,32,31,29);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,25)(24,30)(27,31);(9,11)(10,13)(12,16)(14,15)(17,19,22,18,21,25,26,23)(20,27,29,24,28,31,32,30);(3,8)(5,7)(10,14)(11,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(2,6)(5,7)(10,14)(13,15)(18,23)(19,25)(20,28)(29,32)Polycyclic presentation:
2^9|a^2=i,c^2=g,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7232096);