Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2855 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)(10,16)(12,21)(13,14)(17,19)(18,24)(22,23);(5,6)(7,8)(9,10)(11,12)(13,16)(14,24)(15,18)(17,21)(19,22)(20,23);(2,4)(6,8)(9,11)(10,12)(13,17)(14,19)(15,20)(16,21)(18,23)(22,24);(2,4)(9,12,13,22,15,23,14,21)(10,17,24,20,18,19,16,11);(9,13,15,14)(10,16,18,24)(11,19,20,17)(12,22,23,21)Polycyclic presentation:
2^9|d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72312914, #72313004, #72313010 |
| maximal normal subgroup of | #72312914, #72313004, #72313010 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409327);