Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#3268 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2362 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)(13,14,17,22)(15,24,25,19)(16,28,27,21)(18,20,26,23);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,22,24)(16,18,27,26)(20,28,23,21);(6,9)(10,12)(14,20)(15,25)(18,26)(19,21)(22,23)(24,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,27)(14,21,22,28)(15,26,25,18)(19,23,24,20);(7,11)(10,12)(15,25)(16,27)(19,24)(21,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442814);