Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19)(18,23,25,31)(20,28)(21,29)(22,27)(24,26,30,32);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,27)(23,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=h,c^2=fgh,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72279567, #72280413, #72303170, … |
| maximal normal subgroup of | #72279567, #72280413, #72303170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301231);