Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2063 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)(7,10)(8,11)(9,12)(13,14,18,24,19,23,28,26)(15,25,21,17,22,16,27,20);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,22)(14,20,23,17)(16,26,25,24)(18,27,28,21);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,26)(17,28)(18,20)(19,25)(22,24)(23,27);(2,4)(5,8,9,7)(6,11,12,10)(13,17,19,20)(14,22,23,15)(16,28,25,18)(21,24,27,26);(14,23)(16,25)(17,20)(24,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360365);