Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489701 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,16)(12,15)(13,22)(14,18)(17,20)(19,23)(21,24);(1,2)(3,6)(4,8)(5,7)(10,15)(11,13)(12,20)(16,17)(18,24)(19,21)(22,23);(1,2)(3,6)(4,8)(5,7);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,16,18,22)(12,19,21,20)(15,23,24,17)Polycyclic presentation:
2^9|d^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72321581, #72342601, #72342602, … |
| maximal normal subgroup of | #72321581, #72342601, #72342602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397159);