Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#3106 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(17,24)(20,21)(22,26)(27,28);(2,3)(13,14,18,22)(15,20,23,28)(16,21,25,27)(17,26,24,19);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,17)(18,23)(19,27)(21,26)(22,28)(24,25);(1,3,4,2)(13,16)(14,21)(15,24)(17,23)(18,25)(19,20)(22,27)(26,28);(5,7,8,11)(6,10,9,12)(15,23)(16,25)(20,28)(21,27)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6374428);