Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417907 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15259 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1966 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)(3,8)(4,17,19,28)(5,11)(6,22)(7,12)(9,20,27,21)(10,14)(13,29,30,26)(15,24)(16,25)(18,23,31,32);(1,3)(2,8)(4,18,19,31)(5,15)(6,14)(7,16)(9,26,27,29)(10,22)(11,24)(12,25)(13,20,30,21)(17,32,28,23);(1,4)(2,9)(3,13)(5,21)(6,20)(7,19)(8,23)(10,26)(11,28)(12,27)(14,18)(15,31)(16,30)(17,22)(24,29)(25,32);(1,3)(2,10)(4,13)(5,14)(6,15)(7,16)(8,22)(9,26)(11,25)(12,24)(17,23)(18,21)(19,30)(20,31)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216071, #72216847, … |
| maximal normal subgroup of | #72213514, #72216071, #72216847, … |
| maximal quotient of | #72250582, C2×#417907, #72252189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325103);