Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | #39335 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,22)(15,16,24,27)(17,20,28,25)(18,23,26,21);(5,7)(6,9)(8,12)(10,11)(13,15,18,25,19,24,26,20)(14,17,21,16,22,28,23,27);(2,4)(7,12)(9,10)(13,16)(14,15)(17,26)(18,28)(19,27)(20,23)(21,25)(22,24);(5,7)(6,10)(8,12)(9,11)(13,17,19,28)(14,20,22,25)(15,23,24,21)(16,26,27,18);(7,12)(9,10)(15,24)(16,27)(17,28)(20,25)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504350);