Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13775 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1946 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)(3,11,13,7)(4,15,16,8)(5,10)(6,9)(12,19)(14,24,27,20)(17,28,29,21)(18,22)(23,25)(26,30)(31,32);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31);(7,11)(19,23)(20,24)(30,31);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,15)(10,21)(11,23)(13,25)(14,26)(18,29)(20,30)(22,28)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#355102, #72240453, #72240457, … |
| maximal normal subgroup of | C2×#355102, #72240453, #72240457, … |
| maximal quotient of | #72224724, #72224728, #72224730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262298);