Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,22,17,24,16,23)(15,21,25,19,27,20,26,28);(1,2)(3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,27)(18,26)(20,23)(21,22)(24,28);(2,4)(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,26)(17,27)(18,25)(19,23)(21,24)(22,28);(2,4)(13,15)(14,21)(16,26)(17,27)(18,25)(19,22)(20,24)(23,28);(13,16,17,18)(14,22,24,23)(15,25,27,26)(19,21,28,20)Polycyclic presentation:
2^9|a^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352852);