Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | #32884 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,21)(15,23,16,19)(18,27,26,22)(20,24,28,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,17)(18,24)(20,27)(21,23)(22,28)(25,26);(2,4)(13,16,17,15)(14,20,21,28)(18,24,26,25)(19,22,23,27);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(15,16)(19,23)(20,28)(24,25);(2,4)(15,16)(19,23)(20,28)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484284);