Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,17,14,18,12,16)(11,19,20,15,22,23,21,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24);(3,7)(5,8);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265388, #72265483, #72265491 |
| maximal normal subgroup of | #72265388, #72265483, #72265491 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92578);