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) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3521 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,23)(16,18)(17,22)(19,21)(20,26)(24,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,21,23)(16,17,24,27)(18,25,28,22);(1,3,4,2)(14,21)(18,28)(20,23)(22,25);(13,16)(14,18)(15,17)(19,24)(20,25)(21,28)(22,23)(26,27);(13,17)(14,22)(15,24)(16,26)(18,20)(19,27)(21,25)(23,28);(5,7,8,11)(6,10,9,12)(13,18)(14,16)(15,25)(17,20)(19,28)(21,24)(22,26)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,f^2=gh,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, 7533045);