Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,8,7)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23);(1,2,3,6,5,8,4,7)(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258064, #72258112, #72302146, … |
| maximal normal subgroup of | #72258064, #72258112, #72302146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102168);