Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,23)(15,24,21,20)(16,27,28,26)(17,22,19,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,22)(17,27)(18,21)(19,26)(23,24)(25,28);(2,4)(13,16,19,24,18,28,17,20)(14,21,22,27,23,15,25,26);(2,4)(5,8,9,7)(6,10,11,12)(13,17,18,19)(14,22,23,25)(15,26,21,27)(16,20,28,24);(15,21)(16,28)(20,24)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351100);