Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482709 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17595 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | #60698 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
701 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)(3,14,16,28)(4,19,21,29)(5,12)(6,9)(7,13)(8,15)(10,17,30,18)(11,23,26,24)(20,31)(22,27)(25,32);(1,3)(2,8)(4,20)(5,18)(6,17)(7,16)(9,25)(10,26)(11,30)(12,31)(13,27)(14,19)(15,24)(21,32)(22,23)(28,29);(2,9,13,12)(4,21)(8,25,27,31)(10,28,30,14)(11,19,26,29)(15,22)(20,32)(23,24);(1,3)(2,10,13,30)(4,22)(5,17)(6,18)(7,16)(8,26,27,11)(9,28,12,14)(15,21)(19,25,29,31)(20,23)(24,32)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,f^d=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231615, #72231624, #72231704, … |
| maximal normal subgroup of | #72231615, #72231624, #72231704, … |
| maximal quotient of | C2×#482709, #72240794, #72240795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389905);