Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493492 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3500 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2256 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)(3,11,14,23)(4,15,17,7)(5,9)(6,10)(8,13,22,18)(12,26)(16,30,32,21)(19,24)(20,27)(25,28)(29,31);(2,7)(8,20)(9,21)(10,15)(11,25)(22,29)(23,26)(24,30);(1,3,6,14)(2,8,10,22)(4,12,17,28)(5,18,19,13)(7,20,15,29)(9,23,24,11)(16,31,32,27)(21,26,30,25);(5,19)(8,22)(9,24)(13,18)(16,32)(20,29)(21,30)(27,31)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225826, #72260467, #72260481, … |
| maximal normal subgroup of | #72225826, #72260467, #72260481, … |
| maximal quotient of | #72225792, #72226017, #72232831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400690);