Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | #32913 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 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)(13,14)(15,20)(16,26)(17,24)(18,23)(19,22)(21,27)(25,28);(1,2,3,4)(7,8)(10,11)(13,15)(14,20)(16,19)(17,27)(18,25)(21,24)(22,26)(23,28);(5,7,9,8)(6,10,12,11)(13,16,19,15)(14,21,22,28)(17,27,18,25)(20,23,26,24);(5,8,9,7)(6,11,12,10)(14,22)(20,26)(21,28)(23,24);(6,12)(10,11)(13,17,19,18)(14,23,22,24)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394922);