Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,10,17,7)(6,11)(12,20)(13,21)(14,22,26,18)(15,24)(16,23,28,19)(25,30)(27,31,32,29);(2,7)(4,15)(5,17)(8,18)(9,23)(10,11)(13,25)(14,26)(19,24)(20,22)(21,31)(29,30);(1,3)(2,8)(4,13)(5,14)(6,12)(7,18)(9,21)(10,22)(11,20)(15,25)(16,27)(17,26)(19,29)(23,31)(24,30)(28,32);(3,12)(8,20)(13,25)(14,26)(18,22)(21,30)(27,32)(29,31);(1,4,6,15)(2,9,11,24)(3,13,12,25)(5,16,17,28)(7,19,10,23)(8,21,20,30)(14,27,26,32)(18,29,22,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72261708, #72261739, #72261759, … |
| maximal normal subgroup of | #72261708, #72261739, #72261759, … |
| maximal quotient of | #72318116, C2×#7622469, #72318754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529665);