Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3918 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)(3,5)(4,6)(7,8)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,31,33,26)(21,32,35,27)(23,28)(24,29)(25,30)(34,39,40,37)(36,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,26)(17,27)(18,19)(23,33)(24,34)(25,35)(28,31)(29,37)(30,32)(36,40)(38,39);(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,29)(18,28)(19,31)(21,34)(22,33)(25,36)(27,37)(30,38)(32,39)(35,40);(12,23)(16,28)(20,33)(24,36)(26,31)(29,38)(34,40)(37,39);(2,6)(5,7)(9,13,14,25)(10,17,18,30)(11,21,22,35)(12,24,23,36)(15,27,19,32)(16,29,28,38)(20,34,33,40)(26,37,31,39)Polycyclic presentation:
2^9|a^2=g,e^2=hi,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281484, #72281894, #72318000, … |
| maximal normal subgroup of | #72281484, #72281894, #72318000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250813);