Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | #37272 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2114 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,25,26,21)(16,18,24,20)(17,22,28,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,21)(16,27)(17,20)(18,28)(19,26)(22,24)(23,25);(7,11)(9,12)(13,16,20,14,19,24,18,23)(15,22,17,25,26,27,28,21);(7,11)(9,12)(13,17)(14,22)(15,18)(16,21)(19,28)(20,26)(23,27)(24,25);(6,10)(9,12)(13,18,19,20)(14,16,23,24)(15,17,26,28)(21,22,25,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=i,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=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380416);