Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1992 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)(5,6,10,12)(7,11,9,8)(13,14,17,21)(15,19,22,18)(16,24,26,20)(23,27,25,28);(6,8)(7,9)(11,12)(13,15,17,22)(14,18,21,19)(16,25,26,23)(20,28,24,27);(1,2,3,4)(5,7,10,9)(6,8,12,11)(13,15)(14,19)(16,23)(17,22)(18,21)(20,28)(24,27)(25,26);(5,8)(6,7)(9,12)(10,11)(15,22)(18,19)(23,25)(27,28);(5,7,10,9)(6,11,12,8)(13,16,17,26)(14,20,21,24)(15,23,22,25)(18,27,19,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282991, #72283144, #72283516, … |
| maximal normal subgroup of | #72282991, #72283144, #72283516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7100519);