Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472908 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16841 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,11)(4,15)(5,18)(7,21)(9,26)(12,27)(13,29)(14,19)(16,30)(22,32)(23,31)(25,28);(1,2)(3,12,14,7)(4,8)(5,9)(6,10)(11,27,19,21)(13,23,16,22)(15,24)(17,25)(18,26)(20,28)(29,32,30,31);(1,3)(2,7)(4,16)(5,19)(6,14)(8,23)(9,27)(10,12)(11,18)(13,15)(17,30)(20,29)(21,26)(22,24)(25,32)(28,31);(1,4,5,20,6,15,18,17)(2,8,9,28,10,24,26,25)(3,13,11,29,14,16,19,30)(7,22,21,31,12,23,27,32)Polycyclic presentation:
2^9|b^2=egh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339020, #72339027, #72339034 |
| maximal normal subgroup of | #72339020, #72339027, #72339034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380104);