Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508066 |
| 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 | C22×D4 | C24 |
| prime radical | #3485 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18,21,23)(19,27,29,42)(20,31,33,25)(22,26,35,37)(24,38,36,30)(28,44,32,43)(34,46,48,41)(39,45,40,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(18,23)(22,35)(24,36)(25,31)(27,42)(34,48)(39,40)(43,44);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19)(18,24)(20,32)(21,29)(22,30)(23,36)(25,40)(26,27)(28,33)(31,39)(34,47)(35,38)(37,42)(41,44)(43,46)(45,48);(4,12)(7,14)(10,16)(13,15)(17,20,21,33)(18,25,23,31)(19,28,29,32)(22,34,35,48)(24,39,36,40)(26,41,37,46)(27,43,42,44)(30,45,38,47)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72317670, #72317700, #72317807, … |
| maximal normal subgroup of | #72317670, #72317700, #72317807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415338);