Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,19,23,17,15,18,21)(16,22,26,27,24,20,25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,15)(16,24)(21,23)(25,26);(6,9)(10,12)(13,15,17,14)(16,22,24,20)(18,21,19,23)(25,27,26,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(7,11)(10,12)(16,24)(20,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452229);