Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #37674 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,22)(16,26)(17,21)(18,20)(19,24)(23,28)(25,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,22)(21,23)(26,28);(2,4)(6,9)(10,11)(13,16,26,14,18,21,17,20)(15,23,28,22,24,27,25,19);(5,7,8,12)(6,10,9,11)(14,20)(15,24)(16,21)(25,28);(2,4)(13,17,18,26)(14,21,20,16)(15,25,24,28)(19,23,22,27)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592141);