Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,24)(17,19)(18,22)(21,26)(23,25)(27,28);(2,3)(6,9)(10,11)(14,19)(16,25)(18,20)(21,23)(24,28)(26,27);(1,2,4,3)(13,15,17,22)(14,20,19,18)(16,23,26,28)(21,27,24,25);(1,3,4,2)(13,16,18,25,17,26,20,27)(14,21,22,28,19,24,15,23);(5,7,8,12)(6,10,9,11)(14,19)(15,22)(16,26)(25,27)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7521033);