Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C2 | C2×#1781 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3426 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)(3,11,14,24)(4,8)(5,17,19,9)(6,10)(7,13)(12,25,27,32)(15,28,30,22)(16,23)(18,21)(20,26)(29,31);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,24)(10,21)(11,17)(13,19)(15,29)(16,27)(22,32)(23,31)(25,28)(26,30);(5,19)(9,17)(11,24)(13,18)(15,30)(22,28)(25,32)(26,29);(1,4,6,16)(2,8,10,23)(3,12,14,27)(5,15,19,30)(7,20,21,31)(9,22,17,28)(11,25,24,32)(13,26,18,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72218460, #72233452, #72233453, … |
| maximal normal subgroup of | #72218460, #72233452, #72233453, … |
| maximal quotient of | C2×#493396, C4×#12707, #72266599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400592);