Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| 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 | #37621 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 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)(7,11)(10,12)(13,14)(15,23)(16,25)(17,21)(18,26)(19,20)(22,27)(24,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,20)(18,24)(21,23)(22,28)(25,27);(6,9)(10,12)(13,16,27,23,17,28,18,19)(14,20,26,24,21,15,22,25);(6,9)(10,12)(15,20)(16,28)(19,23)(24,25);(5,7,8,11)(6,10,9,12)(14,21)(15,20)(22,26)(24,25)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401305);