Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C22⋊C42 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2261 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)(3,12,15,8)(4,9)(5,7)(6,11)(10,19)(13,26,28,22)(14,23,30,20)(16,25)(17,24)(18,21)(27,31,29,32);(2,7,11,10)(4,16,17,18)(8,20,12,23)(9,24)(13,27,28,29)(21,25)(22,26)(31,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,20)(9,22)(10,23)(11,12)(16,27)(17,28)(18,29)(19,30)(21,31)(24,26)(25,32);(1,4)(2,9)(3,13)(5,18)(6,17)(7,21)(8,22)(10,25)(11,24)(12,26)(14,29)(15,28)(16,19)(20,31)(23,32)(27,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219684, #72219686, … |
| maximal normal subgroup of | #72215953, #72219684, #72219686, … |
| maximal quotient of | #72342986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263299);