Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,24,27,33,34,32,36,26,29,31,35,25,30,23,28);(5,7,8,13,9,14,15,19)(6,10,12,17,11,16,18,20)(22,27,29,35)(23,31)(24,32)(25,33)(28,30,36,34);(1,3,4,2)(6,11)(10,16)(12,18)(17,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ch,f^a=fi,d^b=dg,e^b=egi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318551, #72318553, #72318563, … |
| maximal normal subgroup of | #72318551, #72318553, #72318563, … |
| maximal quotient of | #72322378, #72322431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36323);