Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62800 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6,7,8)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(1,2)(3,6)(4,8)(5,7)(10,14)(12,19)(15,20)(18,23);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24);(1,3,5,4)(2,6,7,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72306426, #72306429, #72306438 |
| maximal normal subgroup of | #72306426, #72306429, #72306438 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250968);