Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| prime radical | #24236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,22,18,23,26,19)(15,20,24,16,21,25,28,27);(5,6)(7,10)(8,11)(9,12)(14,19)(16,25)(17,26)(20,27)(22,23)(24,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,22)(17,24)(18,21)(19,27)(23,25)(26,28);(5,8,10,9)(6,11,7,12)(13,16,26,20,18,27,17,25)(14,21,19,24,23,15,22,28);(5,9,10,8)(6,11,7,12)(15,21)(16,27)(20,25)(24,28)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106332);