Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 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)(7,11)(10,12)(13,14,19,22)(15,16,25,27)(17,21,20,28)(18,24,26,23);(5,6)(7,10)(8,9)(11,12)(13,15,18,21,19,25,26,28)(14,20,23,16,22,17,24,27);(2,4)(6,9)(10,12)(13,16)(14,15)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28);(6,9)(10,12)(13,17)(14,21)(15,24)(16,26)(18,27)(19,20)(22,28)(23,25);(5,7,8,11)(6,10,9,12)(14,22)(16,27)(17,20)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398997);