Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,11,15,14,19,13,17)(12,21,20,24,22,16,23,18);(1,2)(3,6)(4,5)(7,8)(10,15,19,17)(12,22)(16,24,21,18)(20,23);(1,3,4,7)(2,6,5,8)(10,16)(11,13)(12,20)(15,24)(17,18)(19,21)(22,23);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=eh,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=deh,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72342431, #72342465, #72342579, … |
| maximal normal subgroup of | #72342431, #72342465, #72342579, … |
| maximal quotient of | #72343346, #72343426, #72343618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405430);