Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124937 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,18,13,17,19,24,14,16,20,23,12,15,21,22);(1,3)(2,5)(4,8)(6,7)(10,15)(11,19)(12,13)(16,17)(18,23)(20,21);(2,6)(5,7)(10,16)(11,20)(15,17)(19,21)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219577, #72219580, #72219581, … |
| maximal normal subgroup of | #72219577, #72219580, #72219581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32133);