Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128065 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1401 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)(3,13)(4,17,19,9)(5,11,6,10)(7,12)(8,15,23,20)(14,27,29,32)(16,24)(18,25)(21,26)(22,28)(30,31);(1,3,6,15,7,16,5,20)(2,8,11,24,12,23,10,13)(4,14,18,30,19,29,21,28)(9,22,25,32,17,31,26,27);(1,4)(2,9)(3,14)(5,21)(6,18)(7,19)(8,22)(10,26)(11,25)(12,17)(13,27)(15,30)(16,29)(20,28)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72234495, #72234496, #72234497, … |
| maximal normal subgroup of | #72234495, #72234496, #72234497, … |
| maximal quotient of | #72234506, #72235162, #72235252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35261);