Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346550 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2408 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 2408 conjugacy classes of subgroups — too many to draw. Only its 196 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)(3,7)(4,12)(5,9)(6,10)(8,11)(13,16)(14,15)(22,28)(29,32);(2,7)(3,5)(8,15)(9,10)(11,13)(12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,3,6,5)(2,7,10,9)(4,11,14,13)(8,15,16,12)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,4,6,14)(2,8,10,16)(3,11,5,13)(7,15,9,12)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72268342, #72268357, #72268360, … |
| maximal normal subgroup of | #72268342, #72268357, #72268360, … |
| maximal quotient of | #72282260 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253753);