Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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×#3268 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2767 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)(9,10,14,12)(11,17,21,24)(13,22,23,16)(15,19,18,20);(9,11,14,21)(10,15,12,18)(13,19,23,20)(16,24,22,17);(2,4)(9,12,14,10)(11,18,21,15)(13,22,23,16)(17,20,24,19);(11,19)(13,23)(15,24)(16,22)(17,18)(20,21);(5,7,8,6)(10,12)(15,18)(16,22)(17,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72263909, #72263935, #72263958, … |
| maximal normal subgroup of | #72263909, #72263935, #72263958, … |
| maximal quotient of | #72264240, #72264570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291192);