Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,14)(12,21)(13,17)(15,20)(16,18)(19,23)(22,24);(5,6)(7,8)(9,11,14,10,16,20,15,18)(12,22,24,21,13,19,23,17);(2,4)(6,8)(9,12)(10,17)(11,19)(13,16)(14,24)(15,23)(18,21)(20,22);(5,6)(7,8)(9,13,16,12)(10,17,18,21)(11,19,20,22)(14,23,15,24);(6,8)(9,14,16,15)(10,11,18,20)(12,23,13,24)(17,22,21,19)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72263951, #72263954, #72264061 |
| maximal normal subgroup of | #72263951, #72263954, #72264061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481493);