Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147752 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10991 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,8)(5,10,19,9)(6,11)(7,14,22,29)(13,23)(15,21)(16,30,31,20)(17,24,32,25)(18,26)(27,28);(1,3)(2,7)(4,16)(5,14)(6,15)(8,23)(9,12)(10,21)(11,22)(13,17)(18,31)(19,29)(20,25)(24,30)(26,27)(28,32);(1,4)(2,8)(3,13)(5,17)(6,18)(7,20)(9,24)(10,25)(11,26)(12,27)(14,16)(15,28)(19,32)(21,23)(22,30)(29,31)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72210464, #72238712, #72238713, … |
| maximal normal subgroup of | #72210464, #72238712, #72238713, … |
| maximal quotient of | #72258216, #72258219, #72258273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54948);