Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,17)(5,7)(9,14)(11,12)(15,29)(18,24)(19,20)(23,26)(27,31)(28,32);(1,2)(3,14)(4,18)(5,11)(6,12)(7,8)(9,16)(10,19)(13,26)(15,30)(17,25)(20,24)(21,23)(22,27)(28,29)(31,32);(1,3)(2,9)(4,15)(5,16)(6,13)(7,21)(8,14)(10,22)(11,26)(12,23)(17,31)(18,28)(19,29)(20,27)(24,32)(25,30);(1,4,6,20)(2,10)(3,15,13,27)(5,19,7,17)(8,18)(9,22)(11,24)(12,25)(14,28)(16,29,21,31)(23,30)(26,32)Polycyclic presentation:
2^9|d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216037, #72216071, … |
| maximal normal subgroup of | #72215936, #72216037, #72216071, … |
| maximal quotient of | C2×#427271, #72252188, #72255611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334467);