Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(14,20)(15,25)(17,18)(22,26)(23,24)(27,28);(1,2,3,4)(7,8)(10,12)(13,14,18,23,19,24,17,20)(15,16,25,27,26,21,22,28);(5,7,9,8)(6,10,11,12)(13,15,18,25,19,26,17,22)(14,21,23,28,24,16,20,27);(5,8,9,7)(6,10,11,12)(13,16,19,21)(14,22,24,25)(15,20,26,23)(17,28,18,27);(6,11)(10,12)(13,17,19,18)(14,20,24,23)(15,25,26,22)(16,27,21,28)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390672);