Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,29)(11,20,24,32,25,37,23,16)(12,17,28,34,26,33,27,40)(21,35,39,31,22,36,38,30);(9,11)(10,16,19,32)(12,22)(13,23)(14,24)(15,25)(17,31,33,30)(18,20,29,37)(21,26)(27,39)(28,38)(34,36,40,35);(2,5)(6,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,26)(16,30)(18,34)(19,33)(20,35)(22,25)(23,38)(24,39)(29,40)(31,32)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,31)(18,34)(19,33)(20,36)(21,25)(23,39)(24,38)(29,40)(30,32)(35,37);(12,26)(17,33)(21,22)(27,28)(30,31)(34,40)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72279774, #72279810, #72279883, … |
| maximal normal subgroup of | #72279774, #72279810, #72279883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279674);