Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
581 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 581 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,13,11,14,10,19)(7,15,16,20,8,17,18,12)(21,22)(23,26,29,36)(24,31,33,27)(25,28)(30,34)(32,35);(1,2,3,4)(5,7,11,8)(6,12,14,20)(9,18,10,16)(13,17,19,15)(21,23)(22,26,28,36)(24,32)(25,29)(27,35,31,34)(30,33);(5,8,11,7)(6,12,14,20)(9,18,10,16)(13,15,19,17)(22,27)(23,29)(26,34)(28,31)(30,32)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gi,d^2=gi,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44544);