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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,18,23)(15,24,21,20)(16,27,28,26)(17,22,19,25);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,22)(17,27)(18,21)(19,26)(23,24)(25,28);(7,11)(9,12)(13,16,19,24,18,28,17,20)(14,21,22,27,23,15,25,26);(7,11)(9,12)(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=h,c^2=g,d^2=i,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, 6364369);