Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,14,7,13)(3,15,19,27,20,26,16,10)(4,21,22,12,23,29,5,9)(17,24,18,25,31,28,32,30);(2,9)(3,16)(5,22)(6,7)(10,25)(11,21)(12,14)(13,29)(15,28)(17,18)(19,20)(24,26)(27,30)(31,32);(1,3)(2,10)(4,17)(5,18)(6,16)(7,19)(8,20)(9,24)(11,26)(12,28)(13,15)(14,27)(21,30)(22,32)(23,31)(25,29);(1,4,8,23)(2,11,14,13)(3,17,20,31)(5,7,22,6)(9,21,12,29)(10,26,27,15)(16,18,19,32)(24,30,28,25)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408344);