Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3385 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)(13,14,19,24)(15,21,26,27)(16,20,25,28)(17,22,18,23);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,26)(14,20,24,28)(16,17,25,18)(21,23,27,22);(6,9)(7,11)(13,16)(14,21)(15,17)(18,26)(19,25)(20,23)(22,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,28,27);(6,9)(7,11)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502889);