Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2617 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,13,18,14,16,12,17)(11,19,22,23,21,24,20,15);(1,3,6,2)(4,8,5,7)(9,11)(10,15,16,23)(12,20)(13,22)(14,21)(17,24,18,19);(10,16)(12,13)(15,23)(20,22);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,24,23,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231799, #72231818, #72231819, … |
| maximal normal subgroup of | #72231799, #72231818, #72231819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104748);