Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 875 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,17)(13,16)(14,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,33)(25,27)(32,37)(34,39)(35,40)(36,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,15,19,29)(12,21,24,34)(13,25,26,22)(16,30,18,27)(17,28,33,39)(23,36,37,35)(31,40,32,38);(3,4)(6,8)(10,16)(11,20)(13,26)(15,27)(17,31)(18,19)(21,34)(23,37)(28,38)(29,30)(32,33)(39,40);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,17,19,33)(11,21,20,34)(13,23,26,37)(15,28,29,39)(16,31,18,32)(22,35,25,36)(27,38,30,40)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395128);