Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,18,14,17,13,15)(11,16,20,24,21,23,19,22);(5,6)(7,8)(10,15)(11,19)(12,13)(16,23)(17,18)(20,21);(1,2,3,4)(6,8)(9,11,12,20,14,21,13,19)(10,16,18,24,17,23,15,22);(6,8)(9,11,12,20,14,21,13,19)(10,16,18,24,17,23,15,22);(10,17)(15,18)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312885, #72312984, #72313010 |
| maximal normal subgroup of | #72312885, #72312984, #72313010 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351550);