Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2759 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)(5,6,8,7)(9,10,13,18,14,19,23,15)(11,20,21,12,17,16,24,22);(5,7,8,6)(10,15)(12,16)(13,23)(18,19)(20,22)(21,24);(2,4)(9,11)(10,16)(12,15)(13,21)(14,17)(18,22)(19,20)(23,24);(9,12,13,16,14,22,23,20)(10,17,18,24,19,11,15,21);(11,17)(12,22)(16,20)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72263894, #72263958, #72263982, … |
| maximal normal subgroup of | #72263894, #72263958, #72263982, … |
| maximal quotient of | #72264570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289182);