Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511790 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,27)(14,26)(15,21)(16,22)(17,24)(18,23)(19,28)(20,30)(25,29)(31,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,28)(16,29)(17,30)(18,31)(21,27)(22,32)(23,25)(24,26);(5,17)(13,30)(15,18)(28,31);(1,4,6,16)(2,8,10,22)(3,12,14,29)(5,18,17,15)(7,19,20,32)(9,23,24,21)(11,25,26,27)(13,31,30,28)Polycyclic presentation:
2^9|d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72218460, #72233205, C2×#511792, … |
| maximal normal subgroup of | #72218460, #72233205, C2×#511792, … |
| maximal quotient of | #72233008, #72233147, #72233246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418988);