Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #39471 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2275 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,22,20,17)(16,24,26,19)(23,27,25,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,14,18,20,22,21)(16,25,27,24,26,23,28,19);(2,4)(7,12)(9,10)(13,16)(14,19)(15,23)(17,27)(18,26)(20,25)(21,24)(22,28);(5,7)(6,10)(8,12)(9,11)(16,26)(19,24)(23,25)(27,28);(7,12)(9,10)(16,26)(19,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504003);