Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,23)(18,24)(19,28)(20,25)(21,27);(5,6)(7,11)(8,10)(9,12)(13,15,18,22,20,17,19,27)(14,21,28,23,25,16,24,26);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,24)(17,28)(18,26)(19,23)(20,21)(25,27);(7,12)(9,11)(13,17,20,15)(14,23,25,26)(16,28,21,24)(18,22,19,27);(6,10)(9,11)(13,18,20,19)(14,24,25,28)(15,22,17,27)(16,23,21,26)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910398);