Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,23,18,24,16,22)(15,21,25,28,26,20,27,19);(5,7)(6,9)(8,11)(10,12)(13,15,18,26)(14,19,24,28)(16,25,17,27)(20,22,21,23);(1,2,3,4)(6,10)(7,11)(13,15)(14,20)(16,27)(17,25)(18,26)(19,23)(21,24)(22,28);(7,11)(9,12)(13,15)(14,21)(16,27)(17,25)(18,26)(19,22)(20,24)(23,28);(13,16,18,17)(14,22,24,23)(15,25,26,27)(19,21,28,20)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6366805);