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×#525 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7176 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)(15,18)(16,21)(17,22)(19,20)(23,25)(24,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,18,20,27)(16,17,23,26)(21,22,25,28);(1,3,4,2)(14,20)(18,27)(21,25)(22,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,17)(18,22)(19,23)(20,25)(24,26)(27,28);(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,28)(21,27);(13,18,19,27)(14,15,20,24)(16,22,23,28)(17,25,26,21)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=g,f^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, 7533041);