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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,20)(9,13)(10,15)(11,18)(12,23)(14,22)(17,28)(19,31)(21,25)(24,29)(26,34)(27,35)(30,36)(32,33);(5,7)(6,11,13,26,15,25,14,27)(8,21)(9,19)(10,17)(12,28,29,32,30,16,20,31)(18,24)(22,33)(23,35)(34,36);(1,3,4,2)(7,17)(11,25)(16,28)(18,21)(19,33)(26,27)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72223666, C2×#128528 |
| maximal normal subgroup of | #72223666, C2×#128528 |
| maximal quotient of | C8×#1336, C2×#128528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35724);