Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23);(1,2,3,4)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24);(10,16)(15,17)(18,23)(22,24);(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24);(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532986);