Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188062 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2962 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2962 conjugacy classes of subgroups — too many to draw. Only its 310 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,7,15,20)(3,9,13,24,12,19,26,30)(6,8,16,22,17,18,28,29)(11,23,25,32,14,21,27,31);(2,7)(3,11)(4,15)(8,18)(9,21)(12,14)(13,27)(16,28)(19,23)(24,32)(25,26)(30,31);(2,8)(3,12)(4,16)(7,18)(9,23)(11,14)(13,25)(15,28)(19,21)(24,30)(26,27)(31,32);(1,3,6,14)(2,9,8,21)(4,13,16,27)(5,12,17,11)(7,19,18,23)(10,24,22,31)(15,26,28,25)(20,30,29,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72235013, #72235014, #72235020, … |
| maximal normal subgroup of | #72235013, #72235014, #72235020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95363);