Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15453 |
| 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 | |
1378 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)(3,8,15,24)(4,9)(5,12)(6,21)(7,11)(10,20)(13,23,29,22)(14,31,30,26)(16,25,17,32)(18,28)(19,27);(1,3)(2,8,11,24)(4,14,18,30)(5,16)(6,17)(7,15)(9,26)(10,22)(12,25,21,32)(13,19,29,20)(23,27)(28,31);(1,4)(2,9)(3,13,15,29)(5,19)(6,20)(7,18)(8,22,24,23)(10,21)(11,28)(12,27)(14,17,30,16)(25,26,32,31);(1,4)(2,10)(3,14,15,30)(5,20)(6,19)(7,18)(8,23,24,22)(9,21)(11,27)(12,28)(13,17,29,16)(25,26,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216071, #72216077, #72216166, … |
| maximal normal subgroup of | #72216071, #72216077, #72216166, … |
| maximal quotient of | C2×#432241, #72255844, #72265114 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339437);