Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13867 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3715 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)(3,11,15,7)(4,16,17,8)(5,19)(6,9)(10,21)(12,14,23,20)(13,22)(18,24,30,29)(25,27)(26,28)(31,32);(1,3)(2,7)(4,13)(5,20)(6,15)(8,22)(9,11)(10,12)(14,21)(16,25)(17,27)(18,31)(19,23)(24,26)(28,30)(29,32);(1,2)(3,12)(4,8)(5,19)(6,9)(7,20)(10,21)(11,14)(13,26)(15,23)(16,17)(18,29)(22,31)(24,30)(25,28)(27,32);(1,4)(2,8)(3,13)(5,18)(6,17)(7,22)(9,16)(10,24)(11,25)(12,26)(14,28)(15,27)(19,29)(20,31)(21,30)(23,32)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215925, #72219682, #72219683, … |
| maximal normal subgroup of | #72215925, #72219682, #72219683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266739);