Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3505 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,13)(5,12,16,9)(6,10)(8,15);(1,3,6,14)(2,7,10,11)(4,12,15,9)(5,13,16,8)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,4)(2,8)(3,12)(5,7)(6,15)(9,14)(10,13)(11,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)(19,26)(22,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268335, #72268336, #72268339, … |
| maximal normal subgroup of | #72268335, #72268336, #72268339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174626);