Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,20,26,19,23,18,22)(15,16,21,17,25,28,24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,26)(18,21)(19,25)(20,24)(22,28)(23,27);(2,4)(6,9)(10,12)(13,16)(14,21)(15,22)(17,20)(18,27)(19,28)(23,24)(25,26);(5,7)(6,10)(8,11)(9,12)(13,17)(14,15)(16,18)(19,27)(20,28)(21,26)(22,24)(23,25);(7,11)(10,12)(13,18,19,20)(14,22,23,26)(15,21,25,24)(16,17,28,27)Polycyclic presentation:
2^9|a^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449870);