Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232153 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19745 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
503 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8,11,7)(4,12,14,16)(5,9)(6,10)(13,15)(17,18)(19,24,29,25)(20,30,28,32)(21,23)(22,27)(26,31);(1,3)(2,7)(4,13)(5,14)(6,11)(8,10)(9,12)(15,16)(18,23)(19,28)(20,29)(21,31)(26,27)(30,32);(1,4,13,3,6,14,5,11)(2,8,9,12,10,7,15,16)(17,19,21,20,22,29,31,28)(18,24,23,30,27,25,26,32);(2,9,10,15)(7,16,8,12)(17,20,31,19,22,28,21,29)(18,25,23,32,27,24,26,30)Polycyclic presentation:
2^9|a^2=g,c^2=efh,d^2=ei,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi44 conjugacy classes — view the full character table.
| Fitting subgroup of | #72245040, #72245210, #72245241 |
| maximal subgroup of | #72245040, #72245124, #72245207, … |
| maximal normal subgroup of | #72245040, #72245124, #72245207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139363);