Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3448 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)(9,10,14,18,15,19,13,17)(11,20,21,23,22,24,16,12);(5,6)(7,8)(9,11)(10,12)(13,21)(14,16)(15,22)(17,20)(18,24)(19,23);(2,4)(6,8)(9,12)(10,11)(13,24)(14,20)(15,23)(16,17)(18,21)(19,22);(2,4)(9,12)(10,16)(11,17)(13,20)(14,24)(15,23)(18,22)(19,21);(9,13,15,14)(10,17,19,18)(11,21,22,16)(12,20,23,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,f^b=fi,e^c=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315656, #72315683, #72315774, … |
| maximal normal subgroup of | #72315656, #72315683, #72315774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408197);