Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493653 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63066 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,18,20,29)(5,11,6,12)(7,13)(8,14)(9,17,25,21)(10,16,28,23)(19,32)(22,24)(26,31)(27,30);(1,3,8,7)(2,9,14,25)(4,16,20,23)(5,17,6,21)(10,24,28,32)(11,15,12,13)(18,27,29,26)(19,30,22,31);(1,4)(2,10)(3,16)(5,19)(6,22)(7,23)(8,20)(9,24)(11,26)(12,27)(13,29)(14,28)(15,18)(17,30)(21,31)(25,32);(1,5,8,6)(2,11,14,12)(3,17,7,21)(4,19,20,22)(9,15,25,13)(10,26,28,27)(16,30,23,31)(18,32,29,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72234103, #72260481, #72260499, … |
| maximal normal subgroup of | #72234103, #72260481, #72260499, … |
| maximal quotient of | #72234065, #72260922, #72261111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400852);