Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111253 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,19,22,15)(12,23,24,33)(13,18)(14,17)(16,25,31,26)(20,35)(21,34,39,29)(27,32)(28,37)(30,38)(36,40);(1,2,4,6)(3,8,7,5)(9,11,14,22)(10,15,17,19)(12,20,24,36)(13,21,27,39)(16,30,31,28)(18,29,32,34)(23,35,33,40)(25,37,26,38);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,20)(13,26)(14,24)(15,28)(17,31)(18,33)(19,30)(21,38)(22,36)(23,32)(25,27)(29,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dh,e^c=eh,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18450);