Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,20,15,21,14,19)(11,22,25,34,26,37,24,17)(12,27,29,36,31,35,30,18)(16,28,32,23,33,39,38,40);(1,3,5,6,4,7,8,2)(9,11)(10,16,21,33)(12,28)(13,24)(14,25)(15,26)(17,18,34,36)(19,32,20,38)(22,35,37,27)(23,30)(29,40)(31,39);(9,11)(10,17)(12,23)(13,25)(14,24)(15,26)(16,18)(19,37)(20,22)(21,34)(27,32)(28,30)(29,39)(31,40)(33,36)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fi,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216392, #72216443, #72216448, … |
| maximal normal subgroup of | #72216392, #72216443, #72216448, … |
| maximal quotient of | #72225151, C2×#135832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43028);