Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1626 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1626 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,21,31)(12,22,24,33)(13,26,28,17)(16,27,32,25)(19,34,23,39)(20,36,38,30)(29,37,40,35);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,23,24,19)(13,20)(14,21)(16,29,32,40)(17,30)(18,31)(22,34,33,39)(25,37,27,35)(26,36)(28,38);(5,8)(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,33)(18,32)(20,37)(21,23)(22,26)(25,28)(30,39)(31,40)(34,36)(35,38);(13,28)(17,26)(20,38)(22,33)(25,27)(30,36)(34,39)(35,37)Polycyclic presentation:
2^9|a^2=h,b^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72256098, #72256326, #72256333, … |
| maximal normal subgroup of | #72256098, #72256326, #72256333, … |
| maximal quotient of | #72262567, #72262631, #72262794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159152);