Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | #37301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 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)(7,10)(8,11)(9,12)(14,19)(16,17)(20,25)(21,26)(22,24)(27,28);(1,2,3,4)(7,8)(10,11)(14,20)(19,25)(21,27)(26,28);(5,7,9,8)(6,10,12,11)(13,14)(15,21)(16,19)(17,25)(18,20)(22,28)(23,27)(24,26);(5,8,9,7)(6,11,12,10)(13,15,18,23)(14,21,20,27)(16,24,17,22)(19,26,25,28);(6,12)(10,11)(13,16,18,17)(14,19,20,25)(15,22,23,24)(21,28,27,26)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394945);