Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485343 |
| 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 | |
829 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 829 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)(11,19)(12,16)(13,26)(14,18)(15,22)(17,27)(20,34)(21,33)(23,37)(24,39)(25,32)(28,35)(29,36)(30,38)(31,40);(1,2)(3,6)(4,8)(5,7)(11,20)(15,28)(17,26)(21,23)(22,35)(29,30)(31,39)(36,38);(9,11)(10,15,18,28)(12,23)(13,22)(14,20)(16,30,32,29)(17,19,26,34)(21,25)(24,38)(27,35)(31,37,39,33)(36,40);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,18,32)(11,21,20,23)(13,24,27,40)(15,29,28,30)(17,31,26,39)(19,33,34,37)(22,36,35,38)Polycyclic presentation:
2^9|c^2=hi,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=di,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392673);