Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | #23393 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2386 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)(7,10)(8,9)(11,12)(13,14,18,22,19,23,17,27)(15,24,25,28,16,21,26,20);(1,2)(3,4)(13,15)(14,20)(16,19)(17,25)(18,26)(21,22)(23,28)(24,27);(2,4)(6,9)(10,12)(13,15,19,16)(14,21,23,24)(17,26,18,25)(20,27,28,22);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,21,23,24)(17,25,18,26)(20,27,28,22);(13,17,19,18)(14,22,23,27)(15,25,16,26)(20,21,28,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356193);