Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #3459 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
945 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,21,20,22,24,29,28)(19,23,27,26,25,30,32,31);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(19,25)(23,30)(26,31)(27,32);(17,20,21,24,22,28,29,18)(19,26,27,30,25,31,32,23)Polycyclic presentation:
2^9|a^2=g,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,f^b=fh,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268335, #72268352, #72268355, … |
| maximal normal subgroup of | #72268335, #72268352, #72268355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116025);