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 | #33183 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2130 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)(5,6,9,10)(7,11,8,12)(13,14,19,16)(15,20,24,25)(17,21,27,23)(18,22,28,26);(6,10)(11,12)(13,15,19,24)(14,20,16,25)(17,22,27,26)(18,21,28,23);(2,4)(13,16,19,14)(15,17,24,27)(18,22,28,26)(20,21,25,23);(5,7,9,8)(6,11,10,12)(13,17,19,27)(14,21,16,23)(15,22,24,26)(18,20,28,25);(5,8,9,7)(6,12,10,11)(15,23)(17,20)(18,28)(21,24)(22,26)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250012, #72256943, #72256952 |
| maximal normal subgroup of | #72250012, #72256943, #72256952 |
| maximal quotient of | #72264544, #72264567, #72264568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485531);