Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | #24378 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3196 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)(7,11)(10,12)(13,14,17,23,18,22,26,19)(15,20,24,28,16,21,25,27);(5,6)(7,10)(8,9)(11,12)(14,19)(15,16)(17,26)(20,28)(21,27)(22,23);(2,4)(6,9)(10,12)(13,15,18,16)(14,20,22,21)(17,24,26,25)(19,27,23,28);(6,9)(10,12)(13,16,18,15)(14,21,22,20)(17,25,26,24)(19,28,23,27);(5,7,8,11)(6,10,9,12)(14,22)(15,16)(19,23)(24,25)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398452);