Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | |
3391 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)(11,19)(12,14)(13,17)(15,18)(16,22)(20,21)(23,24);(5,6)(7,8)(9,11,14,23,15,21,13,22)(10,16,17,20,18,24,12,19);(2,4)(6,8)(9,12,13,18,15,17,14,10)(11,20,22,16,21,19,23,24);(6,8)(11,21)(16,24)(19,20)(22,23);(9,13,15,14)(10,17,18,12)(11,22,21,23)(16,20,24,19)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313004, #72313010, #72313076, … |
| maximal normal subgroup of | #72313004, #72313010, #72313076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346160);