Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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×#3366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1979 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)(5,6,8,9)(7,10,11,12)(13,14,17,20)(15,22,16,19)(18,21,25,28)(23,26,24,27);(6,9)(10,12)(13,15)(14,19)(16,17)(18,23)(20,22)(21,27)(24,25)(26,28);(1,2,3,4)(14,20)(15,23)(16,24)(18,25)(19,26)(22,27);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,22)(18,24,25,23)(21,27,28,26);(15,23)(16,24)(18,25)(19,27)(21,28)(22,26)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257277, #72257495, #72263839 |
| maximal normal subgroup of | #72257277, #72257495, #72263839 |
| maximal quotient of | #72265166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490865);