Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | #37639 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 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)(13,14,19,22)(15,17,21,18)(16,20,25,28)(23,24,27,26);(5,6)(7,10)(8,9)(11,12)(13,15,18,14,19,21,17,22)(16,23,26,20,25,27,24,28);(2,4)(6,9)(7,11)(13,16)(14,20)(15,23)(17,24)(18,26)(19,25)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,15)(16,24,25,26)(20,27,28,23);(6,9)(10,12)(16,25)(20,28)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420741);