Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3546 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3512 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)(13,14,18,19,17,22,16,21)(15,23,26,20,24,27,25,28);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(16,18)(20,27)(21,22)(23,28);(6,9)(7,11)(13,15)(14,20)(16,26)(17,24)(18,25)(19,23)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,19)(15,25,24,26)(20,23,28,27);(6,9)(10,12)(13,16,17,18)(14,19,22,21)(15,26,24,25)(20,23,28,27)Polycyclic presentation:
2^9|a^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424021);