Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,12,7,10)(14,19,20,23)(15,16)(17,25)(21,28,27,22)(24,26);(2,4)(6,10,12,11)(13,14,16,21,25,28,15,23,18,20,26,27,17,22,24,19);(14,20)(19,23)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72285924, #72285948, #72285975, … |
| maximal normal subgroup of | #72285924, #72285948, #72285975, … |
| maximal quotient of | #72286517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33982);